Nonparametric Tail Risk, Stock Returns,

and the Macroeconomy

Caio Almeida1, Kym Ardison1, Rene Garcia2, and Jose Vicente3

1EPGE/FGV, Rio de Janeiro, 2Universite de Montreal and Toulouse School of Economics and3IBMEC Business School and Banco Central do Brasil

Address correspondence to Caio Almeida, FGV-EPGE, Praia de Botafogo, 190, Rio de Janeiro, Brazil, or

e-mail: caio.almeida@fgv.br.

Received January 31, 2017; accepted February 7, 2017

Abstract

This paper introduces a new tail-risk measure based on the risk-neutral excess ex-pected shortfall of a cross-section of stock returns. We propose a novel way to riskneutralize the returns without relying on option price information. Empirically, weillustrate our methodology by estimating a tail-risk measure over a long historicalperiod based on a set of size and book-to-market portfolios. We find that a risk pre-mium is associated with long-short strategies with portfolio sorts based on tail-risksensitivities of individual securities. Our tail-risk index also provides meaningful in-formation about future market returns and aggregate macroeconomic conditions.Results are robust to the cross-sectional information and other parameters selectedto compute the tail-risk measure.

Key words: economic predictability, prediction of market returns, risk factor, risk-neutral prob-

ability, tail risk

JEL classification: G12, G13, G17

* We would like to thank David Veredas, Bruno Feunou, Ron Gallant, Cedric Okou, and Marcio Laurini for

useful discussions, and also Tim Bollerslev, Andrew Patton, and participants at the 2015 CIREQ

Montreal Econometrics Conference, 2015 Econometric Society World Congress, 2015 TSE Financial

Econometrics Conference, the First International Workshop in Financial Econometrics in 2013 (Natal),

2014 SoFiE Conference (Toronto), 2014 SoFiE Harvard Summer School, 2014 LAMES (Sao Paulo), 2015

Brazilian Meeting of Finance, 2015 Brazilian Meeting of Econometrics, seminar participants at the finan-

cial econometrics seminar at Duke University and ORFE financial mathematics seminar at Princeton

University. We also thank Viktor Todorov and Bryan Kelly for providing their data sets. We especially

thank Osmani Guillen for his help in the early stages of this project. The first author acknowledges

financial support from CNPq. The second author acknowledges financial support from ANBIMA. The

third author is a research Fellow of CIRANO and CIREQ. The views expressed in the papers are the

authors’ and do not necessarily reflect those of the Central Bank of Brazil.

VC The Author, 2017. Published by Oxford University Press. All rights reserved.

For Permissions, please email: journals.permissions@oup.com

Journal of Financial Econometrics, 2017, Vol. 15, No. 3, 333–376

doi: 10.1093/jjfinec/nbx007

Advance Access Publication Date: 15 March 2017

Invited Article

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Following the 2007–2009 financial crisis, financial researchers as well as regulators devoted

a lot of attention to the measure and analysis of tail risk and systemic risk and to the eco-

nomic consequences of such risks. An important objective of this research agenda is to

develop measures that are able to capture a common risk to many assets or institutions as

opposed to individual risk measures such as Value-at-Risk (VaR) for one particular portfo-

lio or firm. Although systemic risk focuses on the financial institutions, tail risk looks more

generally at the risk provoked by catastrophic events or disasters and affects all firms. It

could be measured with the time series of one firm or one portfolio but rare occurrences of

disasters make this estimation at best very imprecise. Therefore, the most recent approach

to measuring tail risk has been to exploit the richness of the cross-section of returns. By pro-

posing a power law model for the left tail of asset returns, Kelly and Jiang (2014) obtain a

new measure of time-varying tail risk that captures common fluctuations in tail risk among

individual stocks. Allen, Bali, and Tang (2012) measure systemic risk through the 1% VaR

of several tail distributions of the cross-section of returns of financial firms. The main addi-

tional advantage of these measures is the availability of long historical data for equity

returns at the daily frequency. Both studies use the fact that tail risk plays an important role

in explaining risk premia of equities and contains valuable information for predicting future

economic conditions. One shortcoming is that this measure of tail risk is extracted directly

from raw returns without risk neutralization, which is an adjustment of statistical returns

for changes in risk aversion or time preferences or other changes in economic valuations.

Risk-neutralized measures of tail risk have used option prices. Ait-Sahalia and Lo

(1998) and Ait-Sahalia and Lo (2000) introduced an option-based methodology to recover

the risk-neutral densities for the S&P 500 index. In these papers, the authors stress that

risk-neutral information coming from option prices provides an additional, economically

relevant, source of information to estimate tail risk, in their case VaR. Recently, Bollerslev

and Todorov (2011) proposed a model-free index of investors’ fear adopting a database of

intraday prices of futures and the cross-section of S&P 500 options. They show that tail-

risk premia account for a large fraction of expected equity and variance risk premia as

measured by the S&P 500 aggregate portfolio. In a complementary work, Bollerslev,

Todorov, and Xu (2015) provide empirical evidence that the jump risk component of the

variance risk premium is a strong predictor of future market returns. Still relying on S&P

500 index options, based on its empirical distribution, Bali, Cakici, and Chabi-Yo (2011)

propose risk neutral and objective measures of riskiness, and show that they are good pre-

dictors of future market returns. Siriwardane (2013) proposes another interesting method

to estimate the risk of extreme events. He extracts daily measures of market-wide disaster

risk from a cross-section of equity option portfolios (in particular from puts) using a large

number of firms, and show that these measures are useful to predict business cycle variables

and to construct profitable portfolios sorted by disaster risk. A major limitation of these

studies is the short time span for the availability of option and high-frequency data, in the

order of 20 years, especially when the focus is on rare events. Moreover, several markets

around the world do not have well-established and liquid option markets.

We propose an approach based on a cross-section of portfolio returns, together with a

risk neutralization. Our aim is to develop a simple measure based on easily available data

with a long history and applicable to a vast set of asset markets. This disqualifies option pri-

ces used in the previously cited studies. However, we want to reinterpret the insight pro-

vided in Ait-Sahalia and Lo (2000) by eliciting state-price densities with a nonparametric

334 Journal of Financial Econometrics

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methodology developed in Almeida and Garcia (2016). Based on a family of discrepancy

functions, they derive nonparametric stochastic discount factor (SDF) bounds that naturally

generalize variance (Hansen and Jagannathan, 1991), entropy (Backus, Chernov, and Zin,

2014), and higher-moment (Snow, 1991) bounds. Implicit in these derivations is the com-

putation of the SDF itself that provides the risk neutralization. Therefore, our new tail-risk

measure is based on the risk-neutral expected shortfall (ES) of a set of portfolio returns.

Our aggregate tail-risk measure will simply be the average of the individual portfolio ESs.

To assess the empirical relevance of our tail-risk measure, we first look at how well it

matches the main extreme market events and the business cycle during a long sample period

(July 1926 to April 2014). It captures most events, is more volatile around these events, and

appears as strongly counter-cyclical. We also verify that it correlates well with other measures

of tail risk based on high-frequency and option-implied data as well as macroeconomic varia-

bles. Since our measure is based on risk-neutralized returns, it is essential to compare it with a

measure using both returns and option data. We replicate the methodology used in Ait-Sahalia

and Lo (1998, 2000) and Ait-Sahalia and Lo (2000) for the period January 1996–April 2014

and estimate a comparable tail-risk measure. The time series from our own tail-risk measure

based only on the cross-section of returns follows closely their tail-risk measure time series.

These basic properties are therefore reassuring in qualifying our measure as risk-neutral tail risk.

The second empirical test is to establish the existence and measure the magnitude of a

tail-risk premium in equity returns. Our main hypothesis, as in Kelly and Jiang (2014), is

that investors’ marginal utility is increasing in tail risk. Therefore, we explore long-short

portfolios based on tail-risk hedging capacity exposures and show that, even after control-

ling for several benchmark factors, this strategy provides a sizable negative and statistically

significant alpha for both one-month and one-year horizons. The results are also very simi-

lar to the option-based methodology of Ait-Sahalia and Lo (1998) and Ait-Sahalia and Lo

(2000) on the shorter sample.

Next, we provide evidence that our tail-risk measure anticipates stock market move-

ments for intermediate horizons, even after controlling for several known predicting varia-

bles. Our measure has also some predictive power for economic conditions. We use the

predictive regressions framework of Allen, Bali, and Tang (2012) with forecasting horizons

varying from 1 to 12 months. Our proposed tail-risk measure reasonably anticipates future

declines of the main macroeconomic condition indexes. To establish the “channels”

through which tail risk affects the real economy we also consider Bloom (2009) vector

auto-regressive framework. We show that short-term impulse responses to shocks in tail

risk are associated with lower levels of employment and industrial production and are of

greater magnitude than volatility shocks.

In a robustness section, we show how the risk premia and predictive properties vary

with the proposed measure when we change such parameters as the number and nature of

portfolios used to compute the excess ES, the window length, and the VaR threshold as

well as the discrepancy parameter that controls the risk neutralization. Overall our conclu-

sion is that the results are robust in general and most notably when we use different portfo-

lios such as industry portfolios or financial and real sector portfolios.

Brownlees and Engle (2015) also adopted a shortfall measure to assess systemic risk but

proposed a market-based measure of conditional capital shortfall for individual financial

firms (exposed to capital regulation). We differ by considering a risk-neutral measure for a

larger cross-section of returns. Of course, our measure could be specialized to financial

Almeida et al. j Nonparametric Tail Risk and the Macroeconomy 335

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firms to consider systemic risk and not simply tail risk. Our paper is also intimately related

to Adrian and Brunnermeier (2014) by its use of cross-sectional information. However,

they also consider a systemic risk measure that captures financial industry co-movements or

financial contagion. In our methodology, interconnectedness of assets returns comes from

the risk-neutral probability that gives more weight to states of nature where assets in the

chosen panel have lower returns. Finally, this paper also expands the analysis performed by

Bali, Demirtas, and Levy (2009) in several dimensions. First, while their risk measures are

based on objective probabilities, our tail risk exploits the important information in terms of

downside risk embedded in risk neutral probabilities. Also, while Bali, Demirtas, and Levy

(2009) focus mainly on portfolio formation according to downside risk exposures, our

paper puts more emphasis on analyzing the macroeconomic implications of tail risk and the

channels through which it affects the real economy.

The rest of the paper is organized as follows. Section 1 describes how we construct a

risk-neutralized shortfall measure of tail risk based on a panel of asset returns and explains

how it differs from other available tail-risk measures. We compute a long time series of

such tail risk based on a set of size and book-to-market portfolios. Section 2 starts by ana-

lyzing the correlation of our tail-risk measure with the main existing tail and systemic risk

measures and market index returns. We analyze in detail the relationship of the proposed

measure with its option counterpart using Ait-Sahalia and Lo (2000) methodology and

with the Kelly and Jiang (2014) measure. In Section 3, we study empirically the predictive

properties of our tail-risk measure for market returns and macroeconomic activity indica-

tors. Section 4 analyzes the robustness of our tail-risk measure in various dimensions.

Section 5 concludes with a summary and some potential extensions.

1 A Nonparametric Tail-Risk Measure

1.1. Building a Risk-Neutral Excess ES Measure

Historical VaR was considered for a long time as a good tool for managing the risk of a

portfolio. However, given the relative scarcity of extreme events in historical samples, sev-

eral authors proposed to build measures based on risk-neutral probabilities assigned to his-

torical observations. These were generally computed from option prices on the underlying

(generally the market) as in Ait-Sahalia and Lo (2000) and several other papers.1 These

probabilities incorporate economic conditions and the risk attitude of investors and are

therefore more reliable than historical probabilities. However, since prices for liquid

options are not readily available for many assets and countries, a cross-sectional approach

based on returns has been suggested to compute risk associated with tails.2

For both theoretical and empirical reasons, we choose to base our measure on excess ES

instead of VaR.3 A main advantage of using threshold exceedances comes from the

1 See in particular Ait-Sahalia and Lo (1998), Breeden and Litzenberger (1978), Bates (1991),

Rubinstein (1994), and Longstaff (1995).

2 The most recent literature includes Kelly and Jiang (2014); Allen, Bali, and Tang (2012); and Adrian

and Brunnermeier (2014) to cite a few.

3 Basak and Shapiro (2001) provide an equilibrium analysis that puts forward various counter-

intuitive implications of VaR management and show that ES avoids them. In addition, ES is a coher-

ent measure of risk (see Artzner et al., 1999).

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information it contains about the whole tail of the distribution instead of just a point-wise

percentile as VaR.

For a particular asset i, we define the excess ES as follows:

TRi;t ¼ EQðRÞ½ðRi;s � VaRaðRi;sÞÞjðRi;s�VaRaðRi;sÞÞ�; (1)

where t is the time period for which we are calculating the tail risk, s denotes the possible

states of nature, a is the VaR threshold, and QðRÞ indicates the risk-neutral density. Our

aggregate market tail-risk measure will be based on the average of the ES TRi;t of several

portfolios i. Note that in this design we model Q as a function of the observable returns.

Throughout the paper we calculate tail risk at a monthly frequency. Thus, as it will become

clearer further on, the states of nature (s) will be captured by a number of past daily

returns. By keeping this number relatively small our risk measure will react more quickly to

changes in market conditions given that recent cross-sectional values of asset returns will be

used to estimate the risk-neutral density. This is in contrast with usual VaR measures based

on statistical historical properties of asset returns.4

1.2. A Non-Parametric Risk-Neutral Density

To circumvent the problems related to the availability of options prices, we propose to cal-

culate the RND via nonparametric methods. To that end, Hansen and Jagannathan (1991)

seminal paper proposes to minimize a quadratic loss function to estimate a SDF that prices

exactly a set of basis assets. This insightful approach has been used extensively to assess the

adequacy of financial models with a mean–variance SDF frontier. Almeida and Garcia

(2016) have generalized their methodology to frontiers that involve all moments of the

return distributions and use them to gauge models involving tail events such as disaster and

disappointment aversion models. Given a series of assets returns, in an incomplete market

where there are more states of nature than assets, they find a family of SDFs that minimize

convex functions defined in the space of admissible and strictly positive SDFs. These convex

functions measure the distance between an admissible SDF and the constant SDF of a risk-

neutral economy. Assuming a constant short-term rate and homogenous physical probabil-

ities, just as in a VaR historical simulation, we are able to obtain a direct correspondence

between SDFs and RNDs.

Given that RND are the building blocks of our tail-risk procedure we succinctly expose

the methodology adopted and its implications. Let ðX;F ;PÞ be a probability space, and R

denote a K-dimensional random vector on this space representing the returns of K primitive

basis assets. In this static setting, an admissible SDF is a random variable m for which

E(mR) is finite and satisfies the Euler equation:

EðmRÞ ¼ 1K; (2)

where 1K represents a K-dimensional vector of ones.

As in Hansen and Jagannathan (1991), Almeida and Garcia (2016) are interested in the

implications of Equation (2) for the set of existing SDFs. For a sequence of (mt, Rt) that sat-

isfy Equation (2) for all t, and observing a time series fRtgt¼1;:::;T of basis assets returns, we

4 For an interesting exception, see Glasserman (2015) for a data-driven selection of historical stress

test scenarios that are both extreme and plausible based on empirical-likelihood estimators.

Almeida et al. j Nonparametric Tail Risk and the Macroeconomy 337

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assume that the composite process (mt, Rt) is sufficiently regular such that a time series ver-

sion of the law of large numbers applies.5 Therefore, sample moments formed by finite

records of measurable functions of data Rt will converge to population counterparts as the

sample size T becomes large.

Given a sample of basis assets returns, the set of admissible SDFs will depend on the

market structure. The usual case is to have an incomplete market, that is, the number of

states of nature (T) larger than the number of basis assets K. In such case, an infinity of

admissible SDFs will exist, and if there is no in-sample arbitrage on the basis assets payoff

space (see Gospodinov, Kan, and Robotti, 2016), there will exist at least one strictly posi-

tive SDF (see Duffie, 1996). For each strictly positive SDF there will be a corresponding

risk-neutral density.

Given a convex discrepancy (penalty) function / the generalized, in sample, minimum

discrepancy problem proposed by Almeida and Garcia (2016) can be stated as:

mMD ¼ arg minfm1 ;...;mTg

1

T

XT

i¼1

/ðmiÞ

subject to1

T

XT

i¼1

miðRi �1

a1KÞ� ¼ 0K

1

T

XT

i¼1

mi ¼ a

mi� 0ðor mi > 0Þ 8 i

: (3)

In this optimization problem, restrictions to the space of admissible SDFs come directly

from the general discrepancy function /. The conditions Eðm R� 1a 1K

� �Þ ¼ 0K and

E(m)¼ a must be obeyed by any admissible SDF m with mean a. In addition, whenever

there is a strictly positive solution the implied minimum discrepancy SDF is compatible

with absence of arbitrage in an extended economy that considers derivatives over the

underlying basis assets.6 The choice to impose a non-negativity or strict positivity con-

straint in the optimization problem is dictated by the choice of the discrepancy function

/ð:Þ [see Almeida and Garcia (2016) for a detailed analysis].

Despite the straightforward interpretation of the problem in Equation (3), its solution

is not easy given that the number of unknowns is as large as the size of the sample.

Therefore, Almeida and Garcia (2016) show that one can solve an analogous simpler

dual problem:

k ¼ arg supa2R; k2K

a � aþ 1

T

XT

i¼1

/�;þ aþ k0 Ri �1

a1K

� �� �; (4)

5 For instance, stationarity and ergodicity of the process (mt, Rt) are sufficient (see Hansen and

Richards, 1987). In addition, we further assume that all moments of returns R are finite in order to

deal with general entropic measures of distance between pairs of SDFs.

6 It is important to note that the homogeneous probability assumption will not affect the key insights

we derive from this methodology and, if desired, one could also consider a kernel density to model

the physical probabilities without additional complications.

338 Journal of Financial Econometrics

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where K � RK and /�;þ denote the convex conjugate of / restricted to the non-negative (or

strictly positive) real line,

Ri �1

a1K/�;þ ¼ sup

w2½0;1Þ\domain /zw� /ðwÞ: (5)

In this dual problem, k can be interpreted as a vector of K Lagrange multipliers that

comes from the Euler equations for the primitive basis assets in Equation (3). Almeida and

Garcia (2016) specialize the / function to the Cressie-Read family, /cðmÞ ¼mcþ1�acþ1

cðcþ1Þ ; c 2 R. This family captures as particular cases minimizations of variance (Hansen

and Jagannathan, 1991), higher moments (Snow, 1991), and different kinds of entropy

(Stutzer, 1996) estimators. With this family, closed-form formulas are obtained for k and

mMD:

i. if c > 0,

kc ¼ arg supk2KCR

1

T

XT

i¼1

acþ1

cþ 1� 1

cþ 1ac þ ck0 Ri �

1

a1K

� �� �cþ1c

IKCRðRÞðkÞ !

; (6)

where KCR ¼ k 2 RKj8i ¼ 1; . . . ;T ðac þ ck0ðRi � 1

a 1KÞÞ > 0� �

; IAðxÞ ¼ 1; if x 2 A, and 0

otherwise, and:

miMD ¼ a

ac þ ck0c Ri � 1

a 1K

� �� 1c

1T

XT

i¼1ac þ ck

0c Ri �

1

a1K

� �� �1c

IKCRðRÞðkcÞ; (7)

ii. if c�0,

kc ¼ arg supk2KCR

1

T

XT

i¼1

acþ1

cþ 1� 1

cþ 1ac þ ck0 Ri �

1

a1K

� �� �cþ1c

!; (8)

and:

miMD ¼ a

ac þ ck0c Ri � 1

a 1K

� �� 1c

1T

XT

i¼1ac þ ck

0c Ri �

1

a1K

� �� �1c

: (9)

The implied SDF depends directly on the parameter c. Different choices of c will weight

differently higher-order moments of returns (see Almeida and Garcia, 2016). The choice of

c and the robustness of results with respect to its value are therefore important elements of

our empirical strategy. We discuss these aspects in Sections 1.4 and 4, respectively.

To obtain the risk-neutral probabilities associated with each observation interpreted as

a state of nature, we distort the usual 1=T measure by the computed SDF in Equation (9)

adjusted by the interest rate:

pRNi ¼ mið1þ rÞ

T: (10)

Almeida et al. j Nonparametric Tail Risk and the Macroeconomy 339

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We illustrate the distortions in Figure 1 for a five-asset economy that we will make

explicit in the next section. On the horizontal axis, we plot the returns on the optimal port-

folio resulting from the optimization in Equation (8), while the risk-neutral probabilities

are featured on the vertical axis for various values of the parameter c. The different lines

illustrate clearly the fact that for more negative values of c the implied risk-neutral proba-

bilities give more weight to low returns or “bad” states of nature from an investor perspec-

tive. From the linear outcome generated by the unit value (Hansen and Jagannathan,

1991), the convexity increases as c becomes more negative.

1.3. Choosing the Basis Assets and the Estimation Window

As mentioned earlier in this section, our risk-neutral density estimation methodology

assumes an incomplete market framework. Empirically this translates into an upper bound

in the number of cross-sectional assets for a given number of states of nature T. More pre-

cisely, for a given number of days (states of nature) T we must consider a number of assets

K such that K<T. Most papers (Kelly and Jiang, 2014, in particular) compute a tail-risk

measure at a monthly frequency. To make our measure conditional on recent information,

we therefore limit our sample to the former 30 days preceding the last day of each month.7

Otherwise we would be using old information to predict current and future stock market

and economic conditions. Therefore, we cannot adopt a strategy similar to Kelly and Jiang

(2014) who include all securities available in the CRSP data base at the end of each month

and pool all their returns in the former 30 days to compute their tail-risk measure based on

the Hill estimator.8

Figure 1. This figure presents the implied risk-neutral probabilities for a five asset economy and a vari-

ety of c0s.

7 In the robustness section of the paper, Section 4 we report the sensitivity of our empirical results

to the panel length.

8 In Almeida, Ardison, and Garcia (2016), we compute a tail-risk measure using a similar methodology

with individual assets in a high-frequency environment.

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Given an upper bound of 30 days, we need to base our measure on portfolios. But we

can ask ourselves whether we should limit ourselves to the market portfolio or exploit the

cross-sectional information in say the usual 25 size and book-to-market portfolios.9 Bali,

Brown, and Caglayan (2014) argue that previous studies have shown that investors usually

hold portfolios composed by some market proxy (e.g., stock funds) and individual stocks.

In this environment, individual stocks play a crucial role on the total portfolio tail risk.

Empirically, they find evidence that market tail risk is not capable of providing useful infor-

mation for return prediction whereas individual assets tail-risk measures (idiosyncratic tail

risk) is quite successful in predicting market returns. Expanding on this, Kelly and Jiang

(2014) also argues that the whole cross-section of stocks contains valuable information

regarding the tail behavior of the returns. Therefore, it seems that we should rely on as

much cross-sectional information as possible.

However, using the whole set of 25 portfolios as basis assets in our procedure might pro-

duce risk-neutral probabilities that result in sizable pricing errors on those basis assets.10

Recently Kozak, Nagel, and Santosh (2015) show that a SDF based on the first few principal

components of asset returns explains many anomalies. Their argument rests on the fact that

since asset returns have substantial commonality, absence of near-arbitrage opportunities

implies that the SDF depends only on a few sources of return variation. Therefore, instead of

considering the whole cross-section of the 25 size and book-to-market portfolios we extract

the first five principal components from the 30-day panel.11 To make our measure condi-

tional, we compute the principal components for every end-of-month in the sample. Thus, for

each t, we are not using future information in the tail-risk calculation. Table 1 reveals that

these principal components are responsible for much of the portfolio variations. With five

principal components, close to 90% of the variation is explained for the full sample.

1.4. The Choice of cTo understand the effect of c on our tail-risk measure, we derive a Taylor expansion of the

expected value of /ðmÞ ¼ mcþ1�acþ1

ðcðcþ1ÞÞ around the SDF mean a. Noting that /ðaÞ ¼ 0; /0ðmÞ ¼

Table 1. Principal component variance

Principal component 1 2 3 4 5

Variance (cumulative) 0.62 0.76 0.83 0.86 0.89

Notes: The cumulative variance of the first five principal components is reported. The principal component

analysis was performed for the whole sample (07/1926–04/2014). Each month we compute the five principal

components using the daily returns of the last 30 days for the 25 size and book-to-market Fama–French portfo-

lios, and the corresponding cumulative explained variances. The reported figures are the averages of the

monthly figures over the whole sample.

9 For more details about these portfolios, see http://mba.tuck.dartmouth.edu/pages/faculty/ken.

french/index.html (accessed 17 February 2017).

10 We also computed our tail-risk measure for the whole set of 25 Fama–French portfolios. While the

pricing errors where higher, the overall empirical findings were preserved.

11 In our thorough comparison with the Kelly–Jiang measure in Section 2.3, we also compute our

measure with principal components computed from the whole cross-section of returns.

Almeida et al. j Nonparametric Tail Risk and the Macroeconomy 341

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mc

c ; /00ðmÞ ¼ mc�1; /000ðmÞ ¼ ðc� 1Þ mc�2; /0000ðmÞ ¼ ðc� 1Þðc� 2Þmc�3. . ., Taylor expand-

ing / and taking expectations on both sides we obtain12:

Eð/ðmÞÞ ¼ ac�1

2Eðm� aÞ2 þ ðc� 1Þac�2

3!Eðm� aÞ3 þ ðc� 1Þðc� 2Þac�3

4!Eðm� aÞ4 þ � � �

(11)

Two important effects appear regarding the weights attributed to skewness and kurtosis in

this expansion. First, for values of c close to one, both skewness and kurtosis have small

weights when compared with the variance that has a weight equal to one half in the expan-

sion. This implies that discrepancies with values of c close to one do not capture much of

the higher moment activity of pricing kernels. Once we move to more negative values of c

both skewness and kurtosis receive considerable weights in the expansion. The second

important aspect refers to the relative weights that are given to skewness and kurtosis by

different Cressie-Read functions. For �2 < c < 1 note that the absolute weight given to

kurtosis is smaller than the corresponding weight given to skewness. However, for values

of c < �2, kurtosis receives more weight than skewness. In fact all even higher moments

receive more absolute weight than their corresponding odd higher moments in this region

of c.

To choose a base-case value of c, we rely on a comparison based on a disaster model.

In Almeida and Garcia (2016), we derive bounds for the disaster model in Backus,

Chernov, and Zin (2014). In this model the logarithm of consumption growth is given

by:

gtþ1 ¼ gtþ1 þ Jtþ1; (12)

where gtþ1 is the normal component @ðl;r2Þ and Jtþ1 is a Poisson mixture of normals. The

number-of-jumps variable j takes integer values with probabilities e�s sj

j!, where s is the

jump intensity. Conditionally on the number of jumps, Jt is normal:

Jtjj @ðja; jk2Þ: (13)

In this model, the logarithm of the SDF with power utility is:

log mtþ1 ¼ log b� fgtþ1; (14)

where f is the coefficient of relative risk aversion. Therefore, the mean of the SDF is:

a ¼ exp logb� flþ 1

2ðfrÞ2 þ sðe�faþ0:5ðfkÞ2 � 1Þ

�: (15)

The discrepancy bound for the Cressie-Read family is the expectation of /cðmÞ. It can

easily be obtained by taking the expectation of exp ðcþ 1Þlog m, that is:

d ¼ exp ðcþ 1ÞlogðbÞ � flðcþ 1Þ þ 0:5 � ðfrðcþ 1ÞÞ2

þ sðexp � fðcþ 1Þahþ 0:5ðfðcþ 1ÞkÞ2 � 1Þ:(16)

12 Note that all functions in the Cressie-Read family are analytic, that is, their derivative of any order

exists. For this reason, the only condition that is needed for the Taylor expansion to be valid is the

existence of the first four moments of the MD SDF.

342 Journal of Financial Econometrics

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These bounds have a direct link with the measure of entropy used in Backus,

Chernov, and Zin (2014). When c ¼ �1, our discrepancy function is logðaÞ � EðlogðmÞÞ,which corresponds precisely to the entropy of the pricing kernel reported in their

Equation (13).

For these bounds we are computing the Cressie-Read bounds with the returns on the

S&P 500 index and equity options strategies on this index. We use four options portfolios

that consist of highly liquid at-the-money (ATM) and out-of-the-money (OTM) European

call and put options on the S&P 500 composite index trading on the Chicago Mercantile

Exchange.13 We want to illustrate here how the frontiers and the disaster model discrep-

ancy vary with the values of c. For the disaster model we use the parameter values reported

in Table II of Backus, Chernov, and Zin (2014), except for the risk aversion parameter that

we set at 8 to match a slightly higher equity premium. For the values of c, we chose three

values that will illustrate the relation between the bounds and the model. In Figure 2, we

plot the three frontiers corresponding to the three values of c 0, �0.5 and �1. We also plot

in the corresponding colors the points corresponding to the discrepancy of the model at the

mean SDF value 0.9946. For c¼ 0 the model passes quite easily because the weights put on

higher moments for building the frontier are relatively small. On the contrary, for c ¼ �1

the restrictions in the bound are more stringent and the model is far below the frontier. For

the intermediate value of c ¼ �0:5, the model point is above the frontier but still closer

than for the two other values.14 Given this illustrative exercise we will choose c ¼ �0:5 as

our base case.

There are also some statistical arguments for choosing this class of loss functions

Kitamura, Otsu, and Evdokmov (2013) showed that, under some technical assumptions,

the Hellinger (c ¼ �0:5) is the more robust one based on an asymptotic perturbation

criterion.15

Furthermore, if we want a tail-risk measure that incorporates higher-order moments

information we must compute a SDF (and therefore a RND) that also considers this infor-

mation, in contrast to the Hansen and Jagannathan one. For instance, Schneider and

Trojani (2015) argue that investors’ “fear” is an aversion to downside risk, exactly what

we aim at capturing with our methodology. Furthermore, they link the concept of “fear”

with investor prudence,16 which is intimately related to negative skewness aversion.

Therefore, a complete and informative risk measure should be able to reflect the informa-

tional content on higher-order moments.

Therefore for the majority of the empirical section we adopt the Hellinger Tail Risk as

the main benchmark (c ¼ �0:5), but we will still present robustness tests with different val-

ues of c in Section 4.

13 These have been constructed by Agarwal and Naik (2004) to study performance of hedge funds.

14 We can use values close to �0.5 where the model point is event closer to the bound but one has

to be careful because important discontinuities may suddenly occur and the frontier may get

closer to the bound for c ¼ �1.

15 In a recent paper, exploring the non-linearity implied by the Hellinger estimator, Schneider and

Trojani (2015) introduce a class of trading strategies that might be used to trade higher moments

of the returns.

16 Almeida and Garcia (2012, 2016) demonstrated that the methodology we consider to estimate the

SDF is compatible with this concept.

Almeida et al. j Nonparametric Tail Risk and the Macroeconomy 343

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1.5. A First Look at the Tail-Risk Measure over Time

Figure 2 plots the evolution of our tail-risk measure, in blue, and a Hodrick–Prescott fil-

tered version, in red, from July 1926 to April 2014. Our measure is very volatile and

features various peaks often coincident with extreme or significant financial, economic,

or political events. These events have affected negatively equity returns in important

ways. Our tail-risk measure captures most of these huge market drops in U.S. history

and also appears to be more volatile in the periods before and after some of them. Of

course two economic and financial crisis periods figure prominently, the Wall Street

Crash of 1929 followed by the Great Depression of 1929–1939 on the left, the begin-

ning of our sample, and the recent financial crisis of 2008–2010 on the right of the

graph. In between these two major crises, the filtered series shows that the overall level

of the tail risk remained below these extremes, except for occasional spikes that are trig-

gered by some specific events. For these sporadic events the mean reversion is much

stronger, as in the dot-com bubble of the early 2000s or the recent European debt crisis

starting at the end of 2009.

Other tail-risk measures have also captured these types of events but since many of them

rely on option prices their span is much narrower, their sample starting at the beginning of

the 1990s.

Figure 2. This figure plots discrepancy bounds for three Cressie-Read measures with the correspond-

ing disaster-model coordinates. The basis assets are the returns on the S&P 500 index and equity

options strategies on this index.

344 Journal of Financial Econometrics

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2 A Comparison With Other Tail-Risk Measures: Correlations and RiskPremia

Since many tail risk or downside risk measures have been proposed in the last 5 years, we

start by computing simple correlations with several of them over different sample periods,

some based on equity returns starting in the 60s, others using options computed from the

beginning of the 90s. We also consider correlations with market indexes and macroeco-

nomic indicators. We investigate in more depth the relationship of our measure with two

prominent tail-risk measures, one based on option prices using the methodology of Ait-

Sahalia and Lo (2000) and one recently proposed by Kelly and Jiang (2014) that uses the

information in the whole panel of individual stock returns. In this section, we will conduct

the comparison in terms of risk premia associated with portfolios sorted according to the

hedging capacity of individual securities against tail risk. We will extend the comparison in

Section 3 with the predictive properties of the measures for market returns and macroeco-

nomic activity indicators.

2.1. Correlations with Other Indexes

In Table 2, we report the correlation coefficients between our tail-risk measure and other

tail-risk measures as well as financial and macroeconomic indexes: the tail-risk measure of

Bollerslev, Todorov, and Xu (2015) based on option prices; the Kelly and Jiang (2014) tail-

risk measure; the VIX; two stock market indexes; the S&P 500 and the CRSP portfolio; the

systemic risk measure of Allen, Bali, and Tang (2012); and the macroeconomic conditions

index of Bali, Brown, and Caglayan (2014).

The first interesting result is that our tail-risk measure has a noticeable correlation

(0.43) with Bollerslev, Todorov, and Xu (2015) tail-risk measure despite the fact that we

do not include any option returns in the computation of our measure. Another important

measure based on option prices is the VIX. Its correlation with our tail-risk measure is

0.56. This measure is often interpreted as a good indicator for investors’ crash fears. More

generally, since option prices reflect investors’ risk attitudes and economic conditions, it

seems that our risk neutralization goes some distance in capturing this important

Figure 3. This figure plots the tail risk calculated using five principal components extracted from the

25 Fama and French size and book-to-market portfolios for the Hellinger estimated risk-neutral density

(c ¼ �0:5). Principal components were extracted for the last day of each month in our sample using 30

days of past returns on the portfolios.

Almeida et al. j Nonparametric Tail Risk and the Macroeconomy 345

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information for extreme risk. In Section 2.2, we will conduct a thorough comparison

between a nonparametric measure incorporating option prices and our measure based only

on risk-neutralized cross-sectional portfolio returns.

Another tail-risk measure has been proposed recently by Kelly and Jiang (2014), there-

after KJ tail-risk measure. Surprisingly, it is not at all correlated with our tail-risk measure.

From a time series point of view, our measure is volatile and has little persistence, while the

KJ measure is highly persistent. The nature of these measures is therefore very different.

The source of this difference may be due to the amount of information used in constructing

the two measures (the whole cross-section of returns for KJ, principal components of the 25

size and book-to-market portfolios in our measure), the fact that KJ is based on raw returns

while ours is computed with risk-neutralized returns, or other differences regarding the

VaR threshold and the estimating window. We will explore all aspects in a thorough com-

parison in Section 2.3. It should be noted that the KJ measure has a negative correlation of

�0.38 with the VIX and a small and positive correlation of 0.23 with the Bollerslev,

Todorov, and Xu (2015) measure of tail risk.

As it should be expected, our measure is counter-cyclical, with negative correlations of

�0.32 and �0.24 with the S&P 500 and CRSP returns, respectively. In Section 3.1, we will

look more closely at the relation between tail risk and the future aggregate returns. Another

important benchmark is the measure of systemic risk (CATFIN) based on the financial sec-

tor by Allen, Bali, and Tang (2012). The computation of CATFIN is related to our measure

since it is estimated using both VaR and ES methodologies, with nonparametric and para-

metric specifications. The correlation with our tail-risk measure is 0.45, which is reassuring

since returns in the financial sector capture financial crisis risk which underlies a good por-

tion of the movements in a more general tail-risk measure based on all equity returns.

Allen, Bali, and Tang (2012) show that high levels of systemic risk in the banking sector

Table 2. Correlations with other tail-risk measures and financial and macroeconomic indicators

Hellinger S&P 500 CRSP Bloom KJ BTX VIX Macro

Hellinger 1.0000

S&P 500 �0.3210 1.0000

CRSP �0.2433 0.9842 1.0000

Bloom 0.4572 �0.1333 �0.1498 1.0000

KJ �0.0723 0.0854 0.0802 �0.0202 1.0000

BTX 0.4303 �0.1293 �0.1341 0.3726 �0.2289 1.0000

VIX 0.5581 �0.3723 �0.3709 0.9288 �0.3820 0.6625 1.0000

Macro 0.4684 �0.0578 �0.0243 0.5809 �0.2210 0.4395 0.5548 1.0000

CATFIN 0.4507 �0.4146 �0.4385 0.3811 �0.0819 0.5206 0.6463 0.5422

Notes: The table reports the correlation coefficients between Hellinger Tail Risk and other tail measures. CRSP

denotes value-weighted CRSP stock index (07/1926–04/2014); Bloom refers to Bloom (2009) volatility factor

(07/1962–06/2008); KJ refers to Kelly and Jiang (2014) tail-risk index (01/1963–12/2010); BTX refers to

Bollerslev, Todorov, and Xu (2015) tail-risk index (01/19963–08/2013); VIX denotes the CBOE volatility

index (01/1990–04/2014); Macro refers to Bali, Brown, and Caglayan (2014) macroeconomic uncertainty

index (01/1994–12/2013); and CAFTIN denotes Allen, Bali, and Tang (2012) systemic risk measure (01/1973–

12/2012).

346 Journal of Financial Econometrics

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impact macroeconomic conditions. Similarly our tail-risk measure is correlated with macro-

economic indicators of Bloom (2009) and Bali, Brown, and Caglayan (2014) (0.46 and

0.47, respectively). In Section 3.2, we study in detail its relationship with the real economy

through the uncertainty channel. The KJ measure of tail risk exhibits no counter-cyclical

behavior nor does it relate to the macroeconomic indicators of Bloom (2009) and Bali,

Brown, and Caglayan (2014).

These correlations suggest a common component between our tail-risk measure based

on the cross-section of risk-adjusted returns and option-based measures of tails risk meas-

ures. They also point out the distinctive behavior of the KJ tail-risk measure despite the use

of the cross-section of returns. We will probe further the relation between these two sets of

measures in the next two sections.

2.2. A Detailed Comparison with an Option-Based Tail-Risk Measure

We have seen that our tail-risk measure is reasonably well correlated with several tail risk

and systemic risk measures. Among them, we want to probe further the relation with

option-based measures. Since we claim that our procedure produces a risk-neutral measure

of tail risk even though we are using only equity returns, it would be reassuring to show

that the extracted measure has much in common with a measure incorporating option pri-

ces. Our empirical strategy consists in applying the methodology in Ait-Sahalia and Lo

(2000) to estimate the excess ES in Equation (1) using the S&P 500 index options and

returns. Ait-Sahalia and Lo (2000) propose a nonparametric VaR measure that incorpo-

rates economic valuation through the state-price density associated with the underlying

price processes. Ait-Sahalia and Lo (1998) propose to estimate an option-pricing formula

Hð:Þ which has the structure of the Black–Scholes formula but where the volatility is esti-

mated nonparametrically, that is HBSðFt;s;X; s; rt;s; rðX=Ft;s; sÞÞ, where Ft;s is the forward

price of the asset (S&P 500 index) at time t for time-to-maturity s, X is the exercise price, r

denotes the interest rate, and rð:Þ is the volatility function. The estimated option-price func-

tion can be differentiated twice with respect to the strike price X to obtain @2Hð:Þ=@X2,

and then the state-price density by multiplying this second derivative by exp ðrt;ssÞ. Ait-

Sahalia and Lo (2000) used this estimated state-price density to compute an economic VaR

but we can of course use it to compute the excess ES.

The main point of Ait-Sahalia and Lo (2000) was to arrive at a VaR value that is

adjusted for risk and time preferences and for changes in economic conditions. We have the

same objective but we differ from their work in one important aspect. We estimate the risk-

neutral density from a panel of equity returns and distort the objective probabilities through

a minimum-discrepancy procedure. We apply the two methodologies for the period

January 1996 to April 2014. Figure 4 features both tail-risk measures. Despite several meth-

odological and empirical differences between the two measures, Figure 4 reveals that both

time series tend to move together with many coincident peaks and troughs.

To analyze more precisely the similarities between the two measures we conduct an

analysis of implied market premium. We first measure the hedging capacity (or insurance

value) of all individual stocks with code 10-11 in CRSP over the period (i.e., their contem-

poraneous betas with respect to the tail-risk measure, ri;t ¼ ai þ biTRi;t17) and then sort

17 This is to be distinguished from the predictive betas computed in Kelly and Jiang (2014),

ri ;tþ1 ¼ ai þ bi TRi ;t , which measure exposure to tail risk.

Almeida et al. j Nonparametric Tail Risk and the Macroeconomy 347

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these betas into 10 portfolios for the lowest hedging portfolio to the highest one. We then

compute the post-formation returns associated with these portfolios over the next month.

The results are reported in Table 3. There are two panels. The upper panel reports the

option-implied tail risk while the lower one features our tail-risk measure with the

Hellinger discrepancy. For each panel, we report several measures of returns. In the first

line, we present the raw average returns for the 10 portfolios and for the difference between

the highest hedging and the lowest hedging portfolios (last column High-Low). For the

other lines we report the returns after controlling for other factors, namely the three Fama–

French factors, to which we add momentum, liquidity, and volatility one by one.

First, looking at the first line which reports the raw average returns, it is remarkable to

see that the magnitudes of the High-Low column are very similar and share the same signs

and statistical significance. The difference is �3.73 for the option-based measure while it is

�3.06 for the Hellinger measure. This suggests that the crash perceptions embedded in the

option prices are stronger than in our nonparametric distortion. After controlling for the

other factors, all the signs remain negative except when we add volatility. In terms of statis-

tical significance the option-implied high minus low returns are estimated more precisely

than the nonparametric measure, which makes sense since we do not incorporate market

prices of options.

There is evidently some short-run correlation between volatility and tail risk. One can

think of an equilibrium explanation in a consumption-based asset pricing model as in

Farago and Tedongap (2015). They propose an intertemporal equilibrium asset pricing

model featuring disappointment aversion and changing macroeconomic uncertainty (con-

sumption volatility) where besides the market return and market volatility, three

disappointment-related factors are also priced: a disappointment factor, a market downside

Figure 4. This figure plots two series of tail risk. The blue line corresponds to the tail risk computed

from the five principal components extracted monthly from the 25 Fama–French size and book-to-mar-

ket portfolios with the estimated Hellinger risk-neutral density (c ¼ �0:5). The red line features the cor-

responding tail-risk measure based on a state-price density obtained with the Yacine and Lo (1998,

2000) methodology applied to a panel of S&P 500 index options. The sample period ranges from

January 1996 to April 2014.

348 Journal of Financial Econometrics

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Tab

le3.

Op

tio

nv

s.H

ellin

ge

rso

rte

dp

ort

folio

s

Opti

on

implied

tail

risk

Port

folio

Low

2.0

03.0

04.0

05.0

06.0

07.0

08.0

09.0

0H

igh

Hig

h–L

ow

Aver

age

retu

rn4.6

32.1

32.4

81.2

61.7

81.1

70.8

50.9

70.8

10.9

0�

3.7

3

(2.5

7)

(2.2

9)

(2.4

9)

(1.5

4)

(1.8

6)

(1.5

5)

(1.3

3)

(1.6

4)

(1.2

3)

(1.7

2)

(�2.3

5)

FF3

3.5

11.4

71.7

40.5

01.1

50.5

50.3

10.4

00.3

60.5

6�

2.9

6

(2.2

2)

(1.5

3)

(1.7

6)

(0.6

2)

(1.1

4)

(0.7

2)

(0.4

4)

(0.7

1)

(0.5

2)

(1.0

0)

(�2.1

5)

FF3þ

MO

M3.7

21.3

21.5

40.4

70.9

30.4

80.3

00.4

50.3

10.6

4�

3.0

8

(2.0

2)

(1.4

1)

(1.5

9)

(0.6

0)

(1.0

1)

(0.6

5)

(0.4

5)

(0.8

1)

(0.4

8)

(1.1

9)

(�1.8

4)

FF3þ

MO

Mþ

LIQ

4.4

31.3

71.5

90.4

11.0

80.6

00.3

70.4

10.5

00.6

3�

3.8

0

(1.7

2)

(1.2

6)

(1.4

2)

(0.4

6)

(0.9

1)

(0.6

8)

(0.4

9)

(0.6

7)

(0.6

2)

(1.0

3)

(�1.6

0)

FF3þ

MO

Mþ

LIQþ

VO

L�

0.8

90.9

20.8

5�

0.1

9�

0.3

7�

0.2

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31.4

32.3

2

(�0.3

3)

(0.5

1)

(0.4

9)

(�0.1

2)

(�0.2

5)

(�0.1

6)

(0.5

9)

(0.1

2)

(0.0

3)

(1.5

6)

(1.0

2)

Hel

linger

tail

risk

Port

folio

Low

2.0

03.0

04.0

05.0

06.0

07.0

08.0

09.0

0H

igh

Hig

h–L

ow

Aver

age

retu

rn4.4

82.7

12.2

01.2

21.8

80.8

40.7

30.6

30.8

81.4

2�

3.0

6

(2.5

2)

(2.4

4)

(2.2

7)

(1.5

3)

(1.9

6)

(1.2

5)

(1.2

0)

(1.1

2)

(1.5

2)

(2.1

8)

(�2.0

3)

FF3

3.3

71.9

21.4

80.4

91.2

70.3

00.2

20.2

10.3

20.9

7�

2.4

0

(2.1

7)

(1.7

2)

(1.5

2)

(0.6

2)

(1.2

9)

(0.4

3)

(0.3

6)

(0.3

6)

(0.5

4)

(1.3

5)

(�1.8

7)

FF3þ

MO

M3.6

71.5

31.2

80.3

71.1

50.2

60.1

90.1

70.4

81.0

5�

2.6

3

(2.0

4)

(1.4

4)

(1.3

6)

(0.4

8)

(1.2

8)

(0.3

8)

(0.3

3)

(0.3

0)

(0.8

2)

(1.5

1)

(�1.6

7)

FF3þ

MO

Mþ

LIQ

4.3

91.7

41.3

40.3

61.3

50.1

90.1

50.1

90.4

41.2

5�

3.1

4

(1.7

3)

(1.3

4)

(1.2

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(0.4

2)

(1.1

7)

(0.2

5)

(0.2

3)

(0.2

9)

(0.6

6)

(1.5

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(�1.3

6)

FF3þ

MO

Mþ

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(�0.0

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(�0.0

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(0.0

1)

(�0.3

7)

(�0.0

2)

(0.1

4)

(1.1

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(0.7

2)

Note

s:T

he

table

report

sth

ere

turn

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dec

ile

port

folios

of

all

CR

SP

stock

sw

ith

code

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of

Ait

-Sahalia

and

Lo

(2000)

and

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sopti

on

pri

ces.

InPanel

B,

the

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fact

or

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dfr

om

the

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rns

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nci

palco

mponen

tsex

tract

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om

the

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size

and

book-t

o-m

ark

etport

folios

acc

ord

ing

toth

em

inim

um

dis

crep

ancy

pro

cedure

tori

sk-a

dju

stre

turn

s.For

each

month

inour

sam

ple

from

January

1996

toA

pri

l2014,w

eso

rtth

est

ock

sand

track

thei

rre

turn

sone

month

post

-form

ati

on.

Inth

efirs

tline

we

report

the

aver

age

port

folio

retu

rns.

Inth

efo

llow

ing

lines

we

report

the

a0s

of

regre

ssio

ns

wher

ew

eco

ntr

olfo

rth

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ctors

indic

ate

din

the

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est

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stic

sco

mpute

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ith

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are

report

edbet

wee

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eses

.

Almeida et al. j Nonparametric Tail Risk and the Macroeconomy 349

Downloaded from https://academic.oup.com/jfec/article-abstract/15/3/333/3072611by Universite de Montreal useron 19 March 2018

factor, and a volatility downside factor. The last factor represents changes in market vola-

tility in disappointing states and calls for a particular premium. Our linear analysis cannot

disentangle these downside effects, hence the correlation between the volatility factor and

the tail-risk factor.

We believe that these results show that our nonparametric tail-risk measure goes a long

way in capturing the risk adjustment ingrained in option prices. Therefore, it can supple-

ment the option-based tail-risk measures for markets where liquid option markets are not

present and for periods where these derivative securities did not exist but where tail risk

was prominent, such as the Great Depression of 1929–1939. Since our measure is based on

a panel of equity returns we need to understand its relation with the Kelly and Jiang (2014)

tail-risk measure, especially since they appear to be uncorrelated.

2.3. A Detailed Comparison with the Kelly and Jiang (2014) Tail-Risk Measure

Figure 5 plots the tail-risk measure based on our expected excess shortfall methodology

together with the Kelly and Jiang (2014) measure. The two series are quite distinct and

seem to relate negatively when big shocks hit the market. Kelly and Jiang (2014) assume

that the lower tail of any asset return i behaves according to the following law:

PðRi;tþ1 < rjRi;tþ1 < ut andF tÞ ¼r

ut

� ��ai=kt

; (17)

with r < ut < 0. The authors estimate the time-varying tail exponent by the Hill estimator:

kHillt ¼ 1

Kt

XKt

k¼1

lnRk;t

ut; (18)

Figure 5. This figure plots two series of tail risk. The blue line corresponds to the tail risk computed

from the five principal components extracted monthly from the 25 Fama and French size and book-to-

market portfolios with the estimated Hellinger risk-neutral density (c ¼ �0:5). The red line features the

tail-risk measure proposed by Kelly and Jiang (2013) based on a Hill estimator computed from the

whole cross-section of CRSP securities available at each time period. The sample period ranges from

January 1996 to December 2012.

350 Journal of Financial Econometrics

Downloaded from https://academic.oup.com/jfec/article-abstract/15/3/333/3072611by Universite de Montreal useron 19 March 2018

where Rk;t is the kth daily return that falls below an extreme value threshold ut during

month t, and Kt is the total number of such exceedances in the cross-section within

month t. They find that the tail exponent is highly persistent and features a high degree of

commonality across firms.

Although the KJ estimator is based on an assumption about the statistical behavior of

the tail of the cross-section distribution of returns, our estimator relies on a risk adjustment

that distorts the returns of the chosen basis assets according to a set of Euler conditions.

This adds a lot of variability to our tail-risk estimator.

Although our tail-risk factor and the Kelly and Jiang (2014) tail-risk measure are com-

puted from a cross-section of equity returns they differ in several aspects. An important dif-

ference relates to the information used by each method. Each month, Kelly and Jiang

(2014) compute their tail-risk measure based on the whole set of individual raw returns

below a threshold of 5% in the elapsed month. To set ideas if we have 5000 individual

securities, the procedure uses 5% of 100,000 observations (5000 20 days), that is 5000

observations. Our measure starts from a set of 25 portfolios sorted by size and book-to-

market value, collects their returns over the last 30 days (a month and a half) and then

extracts five principal components of these 25 series of 30 days, leaving us with five series

of 30 observations. We then compute our excess ES with the risk-adjusted returns at a level

of 10% for each principal component and take the average of the five shortfall measures.

Therefore, there is a fundamental difference since the Kelly and Jiang (2014) method

extracts a common component among the say 5000 tail observations while our procedure

selects extreme observations after a double reduction in information by first looking at

portfolios and then taking principal components of these portfolios. Is this contraction of

information the main reason for the fundamental difference between the two measures of

tail risk?

To answer this question we start by running our procedure with all individual securities.

We compute the five principal components out of the whole universe of available stocks

every month for a window of 30 days. The robustness of the procedure is evaluated as in

Section 2.2 with the returns of 10 portfolios sorted according to the tail-risk hedging

capacity and the resulting high minus low returns. To benchmark the results obtained with

all securities, we first compute the portfolio returns over the 1926–2014 period with our

original data set based on the 25 size and book-to-market portfolios. In Table 4, we report

in the last column the high minus low returns earned post-formation for a one-month hold-

ing period (Panel A) and for a one-year holding period (Panel B). For the short-period

returns, all differences are negative, of similar magnitude across the various controls for

additional factors and statistically significant except for the last line where we add a volatil-

ity factor. The results are similar for the yearly returns but the difference is now positive

and insignificant when we add the volatility factor. Table 5 contains the results of the same

analysis when the five principal components are computed from the whole universe of equi-

ties available in CRSP. Results for both short and long holding periods are very similar.

Moreover, we have conducted the same analysis directly on the 25 size and book-to-market

portfolios instead of their five principal components and results were also very similar.

Therefore, we can conclude confidently that our methodology is robust to the amount of

cross-sectional information used to build the tail-risk measure.

Another source of divergence could be due to our risk-neutralization. To verify if it is,

we define a standardized tail-risk measure based on the average of the excess ES in

Almeida et al. j Nonparametric Tail Risk and the Macroeconomy 351

Downloaded from https://academic.oup.com/jfec/article-abstract/15/3/333/3072611by Universite de Montreal useron 19 March 2018

Tab

le4.

He

llin

ge

rso

rte

dp

ort

folio

s—2

5si

zea

nd

bo

ok-

to-m

ark

et

po

rtfo

lio

s

Panel

A:O

ne-

month

hold

ing

per

iod

Port

folio

Low

2.0

03.0

04.0

05.0

06.0

07.0

08.0

09.0

0H

igh

Hig

h–L

ow

Aver

age

retu

rn2.3

41.3

81.0

40.7

40.8

80.5

60.4

80.3

80.5

30.8

2�

1.5

2

(4.1

8)

(3.5

7)

(3.1

3)

(2.5

1)

(2.8

3)

(2.1

6)

(1.9

9)

(1.7

4)

(2.3

9)

(3.3

0)

(�3.4

1)

FF3

1.2

10.3

10.0

5�

0.2

1�

0.0

8�

0.2

8�

0.3

1�

0.3

5�

0.1

80.0

6�

1.1

5

(3.2

0)

(1.8

5)

(0.3

9)

(�2.1

3)

(�0.6

6)

(�2.9

0)

(�3.3

5)

(�4.3

4)

(�1.5

7)

(0.4

6)

(�3.0

9)

FF3þ

MO

M1.2

40.3

20.0

7�

0.1

9�

0.0

5�

0.2

7�

0.3

0�

0.3

3�

0.1

70.0

5�

1.2

0

(3.1

7)

(1.9

6)

(0.5

1)

(�1.9

6)

(�0.4

1)

(�2.7

5)

(�3.2

2)

(�4.0

5)

(�1.5

3)

(0.3

4)

(�3.1

0)

FF3þ

MO

Mþ

LIQ

1.3

50.3

20.0

7�

0.1

8�

0.0

5�

0.2

7�

0.2

8�

0.3

2�

0.1

60.0

6�

1.2

8

(2.9

0)

(2.0

3)

(0.5

4)

(�1.8

7)

(�0.4

1)

(�2.7

2)

(�3.0

4)

(�3.8

8)

(�1.3

6)

(0.4

8)

(�2.7

9)

FF3þ

MO

Mþ

LIQþ

VO

L0.8

7�

0.0

2�

0.0

5�

0.3

1�

0.1

4�

0.2

1�

0.3

4�

0.3

1�

0.0

80.0

4�

0.8

3

(2.0

2)

(�0.1

3)

(�0.3

2)

(�2.6

2)

(�1.0

3)

(�1.9

8)

(�3.0

0)

(�3.2

0)

(�0.5

3)

(0.2

5)

(�1.8

0)

Panel

B:O

ne

yea

rhold

ing

per

iod

Port

folio

Low

2.0

03.0

04.0

05.0

06.0

07.0

08.0

09.0

0H

igh

Hig

h–L

ow

Aver

age

retu

rn20.8

311.1

99.3

36.0

96.0

64.5

53.5

73.0

44.6

06.9

9�

13.8

4

(3.4

9)

(3.0

4)

(2.8

2)

(2.2

9)

(2.4

3)

(1.8

7)

(1.5

5)

(1.4

4)

(2.0

4)

(2.5

7)

(�3.1

0)

FF3

0.7

7�

3.5

1�

4.0

0�

5.6

2�

4.9

1�

5.9

0�

6.6

1�

6.0

6�

4.8

2�

2.4

0�

3.1

7

(0.2

5)

(�2.1

8)

(�2.6

1)

(�4.6

4)

(�4.0

9)

(�4.3

2)

(�5.0

5)

(�5.1

4)

(�3.7

9)

(�1.7

0)

(�1.1

6)

FF3þ

MO

M2.6

0�

2.3

2�

3.3

4�

5.1

5�

4.8

9�

5.4

0�

6.1

0�

5.3

4�

4.1

1�

2.3

9�

4.9

9

(0.9

3)

(�1.5

3)

(�2.2

6)

(�4.6

8)

(�4.4

2)

(�4.2

7)

(�4.9

9)

(�4.8

7)

(�3.2

3)

(�1.6

4)

(�2.0

1)

FF3þ

MO

Mþ

LIQ

4.0

4�

2.2

6�

3.3

9�

5.2

3�

5.0

4�

5.2

5�

5.9

4�

5.0

8�

3.5

2�

0.9

4�

4.9

8

(1.4

9)

(�1.8

1)

(�2.3

6)

(�5.3

4)

(�4.6

0)

(�4.0

4)

(�4.5

8)

(�4.3

5)

(�2.5

6)

(�0.5

3)

(�2.0

5)

FF3þ

MO

Mþ

LIQþ

VO

L�

15.3

4�

10.2

5�

10.6

1�

10.5

4�

11.1

4�

9.0

2�

8.5

2�

6.7

7�

8.3

5�

3.7

911.5

6

(�1.3

8)

(�1.8

9)

(�2.6

4)

(�3.2

6)

(�4.6

0)

(�3.8

2)

(�3.8

1)

(�3.5

8)

(�4.4

1)

(�1.3

3)

(1.0

3)

Note

s:R

eturn

sare

report

edfo

rdec

ile

port

folios

of

CR

SP

stock

sw

ith

code

10-1

1so

rted

acc

ord

ing

toth

eir

tail-r

isk

hed

gin

gca

paci

ty.

The

tail-r

isk

fact

or

isco

mpute

dfr

om

the

retu

rns

of

the

firs

tfive

pri

nci

pal

com

ponen

tsex

tract

edfr

om

the

25

size

and

book-t

o-m

ark

etport

folios.

The

retu

rns

are

adju

sted

acc

ord

ing

toth

em

inim

um

dis

crep

ancy

pro

cedure

wit

ha

gam

ma

of�

0.5

(Hel

linger

mea

sure

).For

each

month

inour

sam

ple

from

Feb

ruary

1967

toD

ecem

ber

2013,

we

sort

the

stock

sand

track

thei

rre

turn

sone-

month

post

form

ati

on

(Panel

A)

or

one-

yea

rpost

form

ati

on

(Panel

B).

Inth

efirs

tline

we

report

the

aver

age

port

folio

retu

rns.

Inth

efo

llow

ing

lines

we

report

the

a0s

of

regre

ssio

ns

wher

ew

eco

ntr

ol

for

the

fact

ors

indic

ate

din

the

firs

tco

lum

n.N

ewey

–W

est

t-st

ati

stic

sre

port

edbet

wee

npare

nth

eses

are

com

pute

dw

ith

one

lag

for

month

lyre

sult

sand

12

lags

for

yea

rly

resu

lts.

352 Journal of Financial Econometrics

Downloaded from https://academic.oup.com/jfec/article-abstract/15/3/333/3072611by Universite de Montreal useron 19 March 2018

Tab

le5.

He

llin

ge

rso

rte

dp

ort

folio

s—a

llse

curi

tie

s

Panel

A:O

ne

month

hold

ing

per

iod

Port

folio

Low

2.0

03.0

04.0

05.0

06.0

07.0

08.0

09.0

0H

igh

Hig

h–L

ow

Aver

age

retu

rn2.2

41.1

21.0

20.9

30.6

20.6

60.5

40.6

00.4

50.9

7�

1.2

8

(4.0

9)

(3.2

1)

(3.3

1)

(3.0

8)

(2.3

5)

(2.3

9)

(2.2

5)

(2.2

1)

(2.0

9)

(3.3

8)

(�2.9

8)

FF3

1.1

30.0

80.0

40.0

2�

0.2

4�

0.2

0�

0.2

5�

0.2

4�

0.2

50.1

4�

0.9

8

(3.0

2)

(0.6

2)

(0.3

6)

(0.1

4)

(�2.3

1)

(�1.9

7)

(�2.8

5)

(�2.2

8)

(�2.8

3)

(0.9

2)

(�2.5

9)

FF3þ

MO

M1.1

60.1

10.0

70.0

3�

0.2

3�

0.1

8�

0.2

2�

0.2

2�

0.2

60.1

1�

1.0

5

(3.0

1)

(0.7

8)

(0.6

1)

(0.2

6)

(�2.1

6)

(�1.7

3)

(�2.5

8)

(�1.9

9)

(�2.9

6)

(0.7

3)

(�2.6

9)

FF3þ

MO

Mþ

LIQ

1.2

50.1

10.0

80.0

4�

0.2

3�

0.1

8�

0.2

2�

0.1

9�

0.2

50.1

2�

1.1

3

(2.7

2)

(0.8

1)

(0.6

9)

(0.3

3)

(�2.3

0)

(�1.7

1)

(�2.5

7)

(�1.5

8)

(�2.8

4)

(0.7

7)

(�2.4

2)

FF3þ

MO

Mþ

LIQþ

VO

L0.9

0�

0.0

4�

0.1

3�

0.1

7�

0.3

2�

0.2

6�

0.2

1�

0.1

6�

0.2

90.1

3�

0.7

7

(2.2

1)

(�0.2

5)

(�0.8

6)

(�1.0

9)

(�2.7

3)

(�2.5

8)

(�2.0

1)

(�1.3

2)

(�2.7

8)

(0.6

5)

(�1.7

3)

Panel

B:O

ne

yea

rhold

ing

per

iod

Port

folio

Low

2.0

03.0

04.0

05.0

06.0

07.0

08.0

09.0

0H

igh

Hig

h–L

ow

Aver

age

retu

rn18.9

310.3

38.1

86.2

65.6

44.6

44.5

64.3

84.2

69.0

5�

9.8

8

(3.7

0)

(2.9

8)

(2.7

3)

(2.2

8)

(2.0

5)

(1.8

5)

(1.9

0)

(1.8

5)

(1.9

4)

(2.8

7)

(�2.9

2)

FF3

1.9

0�

4.3

3�

4.7

0�

5.8

2�

5.6

8�

5.8

0�

5.6

3�

5.8

2�

4.8

6�

2.3

0�

4.2

0

(0.6

4)

(�2.8

8)

(�3.5

2)

(�4.6

8)

(�3.4

6)

(�4.1

7)

(�4.1

1)

(�4.5

1)

(�3.5

4)

(�1.3

3)

(�1.3

1)

FF3þ

MO

M1.5

4�

3.2

6�

3.9

4�

5.3

4�

4.4

6�

5.1

3�

5.0

4�

5.1

7�

3.9

9�

1.6

7�

3.2

1

(0.6

7)

(�2.4

1)

(�3.2

2)

(�4.5

9)

(�2.8

6)

(�3.6

9)

(�3.7

0)

(�4.1

3)

(�3.0

0)

(�0.9

0)

(�1.3

2)

FF3þ

MO

Mþ

LIQ

2.9

1�

3.1

3�

3.6

1�

4.6

8�

4.7

1�

5.4

9�

5.1

5�

4.8

9�

3.6

6�

0.2

0�

3.1

0

(1.1

8)

(�2.4

9)

(�2.6

9)

(�3.9

2)

(�2.8

9)

(�4.1

2)

(�3.7

3)

(�3.9

5)

(�2.8

1)

(�0.1

0)

(�1.1

6)

FF3þ

MO

Mþ

LIQþ

VO

L�

11.3

4�

10.1

5�

9.5

6�

10.1

8�

11.5

8�

8.2

1�

8.0

4�

9.8

1�

9.9

5�

5.5

05.8

5

(�1.6

1)

(�2.4

1)

(�3.0

4)

(�3.4

3)

(�3.4

2)

(�3.3

3)

(�3.3

6)

(�4.2

3)

(�4.7

7)

(�1.3

0)

(1.0

8)

Note

s:R

eturn

sare

report

edfo

rdec

ile

port

folios

of

CR

SP

stock

sw

ith

code

10-1

1so

rted

acc

ord

ing

toth

eir

tail-r

isk

hed

gin

gca

paci

ty.

The

tail-r

isk

fact

or

isco

mpute

dfr

om

the

retu

rns

of

the

firs

tfive

pri

nci

pal

com

ponen

tsex

tract

edfr

om

the

whole

set

of

secu

riti

esavailable

at

any

poin

tin

tim

e.T

he

retu

rns

are

adju

sted

acc

ord

ing

toth

em

inim

um

dis

crep

ancy

pro

ce-

dure

wit

ha

gam

ma

of�

0.5

(Hel

linger

mea

sure

).For

each

month

inour

sam

ple

from

Feb

ruary

1967

toD

ecem

ber

2013,w

eso

rtth

est

ock

sand

track

thei

rre

turn

sone-

month

post

for-

mati

on

(Panel

A)

or

one-

yea

rpost

form

ati

on

(Panel

B).

Inth

efirs

tline

we

report

the

aver

age

port

folio

retu

rns.

Inth

efo

llow

ing

lines

we

report

the

a0s

of

regre

ssio

ns

wher

ew

eco

ntr

ol

for

the

fact

ors

indic

ate

din

the

firs

tco

lum

n.N

ewey

–W

est

t-st

ati

stic

sre

port

edbet

wee

npare

nth

eses

are

com

pute

dw

ith

one

lag

for

month

lyre

sult

sand

12

lags

for

yea

rly

resu

lts.

Almeida et al. j Nonparametric Tail Risk and the Macroeconomy 353

Downloaded from https://academic.oup.com/jfec/article-abstract/15/3/333/3072611by Universite de Montreal useron 19 March 2018

Equation (1) for the raw returns of the five principal components, a so-called objective

counterpart to our risk-neutral measure. We proceed similarly to sort the cross-section of

returns in 10 portfolios according to their standardized tail-risk hedging sensitivity and

compute the post-formation returns of the portfolios as before. Table 6 presents the results

with this alternative objective tail-risk measure. We note that both in terms of magnitude

and statistical significance results are very different from our reported figures in Table 4,

providing some evidence of the important contribution of the risk-neutralization

procedure.

We conclude this section by some interesting observations about the Kelly and Jiang

(2014) tail-risk measure. If we express the law of tail returns in terms of gross returns

instead of net returns, that is 1þ Rk;t; 1þ r and 1þ u, then the Hill estimator will be

approximately, for small daily returns:

kHillt ¼ 1

Kt

XKt

k¼1

½Rk;t � ut�; (19)

Based on this new Hill estimator, we compute the tail-risk equivalent KJ series with the

whole cross-section at any point in time. We plot in Figure 6 this series together with our

Hellinger measure of tail risk based on the five principal components of the whole cross-

section of returns. Despite the different methodologies and the risk-neutralization, the two

series behave much more similarly than before. Averaging the difference of the logarithms

of the returns or the difference of the returns makes a huge difference on the property of the

tail-risk series. Of course this new Hill estimator is closer to our excess ES measure, the

main difference being the fact that our measure is based on risk-adjusted returns of more

aggregated information. The graphs suggest that these two elements, risk neutralization,

and aggregation of information, are important for the magnitude of the fluctuations in both

series, sometimes enhancing them sometimes dampening them. There is not a clear consis-

tent pattern over time.

3 Predictive Properties of Tail Risk

Predictability of market returns or other financial and macroeconomic variables is another

way to assess the usefulness of our tail-risk measure. A key recent paper by Welch and

Goyal (2008) studies comprehensively the predictive power of variables that have been sug-

gested by the academic literature to be good predictors of the equity premium. Their con-

clusions are rather negative both in-sample and out-of-sample. They find that models have

a poor and unstable predictive performance. Therefore, we want to submit our measure to

the same type of analysis and determine whether it adds to the main predictors established

in the literature. We explore in turn market returns, macroeconomic responses to tail-risk

shocks, and finally how tail-risk anticipates business cycle fluctuations.

3.1. Stock Markets Returns

We provide two sets of predictive regressions. First, in Table 7, we consider simple regres-

sions where we control for market returns (12 lags): r½t;tþk� ¼ aþ bTRt þ cr½t�k;t� þ ut,

where TRt denotes the tail-risk measure. Our goal is to compare the predictive power of

our measure relative to the other proposed tail-risk measures. This has two advantages. It is

354 Journal of Financial Econometrics

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Tab

le6.

Ta

il-r

isk

me

asu

reu

nd

er

ob

ject

ive

ex

cess

sho

rtfa

ll—

sort

ed

po

rtfo

lio

s Panel

A:O

ne-

month

hold

ing

per

iod

Port

folio

Low

23

45

67

89

Hig

hH

igh–L

ow

Aver

age

retu

rn1.8

41.3

71.1

10.7

70.5

90.6

10.6

50.4

90.5

41.1

7�

0.6

7

(4.2

6)

(2.9

1)

(3.2

5)

(2.5

9)

(2.2

8)

(2.4

6)

(2.6

7)

(2.0

5)

(2.2

6)

(3.8

5)

(�2.5

0)

FF3

0.7

40.3

60.1

2�

0.1

6�

0.2

5�

0.1

8�

0.1

3�

0.3

0�

0.2

50.2

8�

0.4

6

(3.5

3)

(1.0

6)

(0.8

0)

(�1.4

2)

(�2.6

8)

(�1.7

3)

(�1.3

5)

(�3.7

1)

(�2.5

8)

(1.7

9)

(�1.9

5)

FF3þ

MO

M0.7

40.3

80.1

3�

0.1

5�

0.2

4�

0.1

7�

0.1

0�

0.2

8�

0.2

30.2

9�

0.4

6

(3.6

1)

(1.0

9)

(0.8

5)

(�1.3

2)

(�2.4

6)

(�1.5

3)

(�1.0

6)

(�3.4

6)

(�2.4

0)

(1.8

3)

(�1.9

9)

FF3þ

MO

Mþ

LIQ

0.7

50.4

60.1

4�

0.1

4�

0.2

2�

0.1

5�

0.1

2�

0.2

7�

0.2

30.3

3�

0.4

2

(3.6

5)

(1.0

8)

(0.9

2)

(�1.2

1)

(�2.3

0)

(�1.2

7)

(�1.3

6)

(�3.3

6)

(�2.3

6)

(2.0

2)

(�1.8

1)

FF3þ

MO

Mþ

LIQþ

VO

L0.3

00.0

60.0

4�

0.2

5�

0.2

6�

0.2

1�

0.1

6�

0.3

4�

0.1

70.4

30.1

2

(1.3

0)

(0.1

8)

(0.2

8)

(�2.1

5)

(�2.1

3)

(�1.3

3)

(�1.3

0)

(�3.6

3)

(�1.3

1)

(2.1

7)

(0.4

6)

Panel

B:O

ne

yea

rhold

ing

per

iod

Port

folio

Low

23

45

67

89

Hig

hH

igh–L

ow

Aver

age

retu

rn17.9

710.1

78.4

95.9

94.5

64.8

94.1

04.1

15.4

610.5

0�

7.4

7

(3.4

7)

(3.0

0)

(2.8

3)

(2.1

8)

(1.8

8)

(1.9

3)

(1.8

4)

(1.8

9)

(2.2

4)

(3.2

0)

(�2.3

8)

FF3

�0.5

1�

3.3

9�

4.0

1�

5.8

7�

5.9

8�

5.5

2�

5.6

8�

5.6

1�

4.7

7�

1.7

0�

1.1

9

(�0.1

8)

(�2.0

2)

(�2.4

9)

(�4.4

6)

(�4.4

5)

(�4.0

4)

(�5.0

7)

(�5.6

0)

(�4.3

7)

(�1.0

5)

(�0.4

5)

FF3þ

MO

M�

0.1

0�

2.2

1�

3.3

5�

5.3

9�

5.4

5�

5.0

9�

5.0

3�

5.0

7�

4.4

8�

0.2

8�

0.1

7

(�0.0

4)

(�1.4

0)

(�2.2

4)

(�4.2

8)

(�4.0

8)

(�4.1

6)

(�4.8

7)

(�5.5

1)

(�4.1

8)

(�0.1

8)

(�0.0

9)

FF3þ

MO

Mþ

LIQ

0.6

2�

1.6

9�

3.0

0�

5.3

5�

5.6

7�

4.7

0�

5.3

0�

5.0

7�

3.8

41.4

10.7

9

(0.2

7)

(�1.0

2)

(�1.8

8)

(�4.1

6)

(�4.1

1)

(�3.4

3)

(�5.5

6)

(�5.4

8)

(�3.4

9)

(0.8

5)

(0.4

2)

FF3þ

MO

Mþ

LIQþ

VO

L�

18.7

4�

13.2

0�

10.4

7�

11.2

7�

8.8

2�

7.9

8�

7.4

1�

8.0

3�

6.2

7�

2.1

516.5

9

(�1.9

3)

(�2.1

9)

(�2.5

5)

(�3.5

6)

(�3.9

5)

(�2.8

7)

(�4.0

2)

(�4.6

6)

(�3.5

9)

(�0.8

8)

(1.7

5)

Note

s:R

eturn

sar

ere

port

edfo

rdec

ile

port

folios

ofC

RSP

stock

sw

ith

code

10-1

1so

rted

acco

rdin

gto

thei

rta

il-r

isk

hed

ging

capac

ity

wit

hre

spec

tto

the

rati

obet

wee

nth

eri

sk-n

eutr

alan

dobje

c-

tive

tail-r

isk

mea

sure

s.T

he

tail-r

isk

fact

or

isco

mpute

dfr

om

the

retu

rns

of

the

firs

tfive

pri

nci

pal

com

ponen

tsex

trac

ted

from

the

25

size

and

book-t

o-m

arket

port

folios.

The

obje

ctiv

eta

il-r

isk

ES

isca

lcula

ted

assu

min

ga

hom

oge

neo

us

pro

bab

ilit

yden

sity

funct

ion.For

each

month

inour

sam

ple

from

Feb

ruar

y1967

toD

ecem

ber

2013,w

eso

rtth

est

ock

san

dtr

ack

thei

rre

turn

sone-

month

post

form

atio

n(P

anel

A)

or

one-

year

post

form

atio

n(P

anel

B).

Inth

efirs

tline

we

report

the

aver

age

port

folio

retu

rns.

Inth

efo

llow

ing

lines

we

report

the

a0s

of

regr

essi

ons

wher

ew

e

contr

olfo

rth

efa

ctors

indic

ated

inth

efirs

tco

lum

n.N

ewey

–Wes

tt-

stat

isti

csre

port

edbet

wee

npar

enth

eses

are

com

pute

dw

ith

one

lag

for

month

lyre

sults

and

12

lags

for

year

lyre

sult

s.

Almeida et al. j Nonparametric Tail Risk and the Macroeconomy 355

Downloaded from https://academic.oup.com/jfec/article-abstract/15/3/333/3072611by Universite de Montreal useron 19 March 2018

a test with respect to the direct competition but it is also a robustness test with respect to

sub-samples of the long sample 1926–2014 since the various tail-risk measures are available

over different periods. We choose the CRSP value-weighted market returns index as our

target.18 Overall, we note that the estimated regression coefficients are positive, meaning

that increases in current tail risk are associated with future higher market returns. Also, for

most of the samples over which we perform the regressions the estimated coefficients are

statistically significant for an interval of 2–6 months. The most robust performance comes

from the long horizons starting in the 1960s. This is not the case of the other tail-risk meas-

ures. The CATFIN index does not show any forecasting power while the Bloom volatility

index is forecasting at a very short horizon of 1 or 2 months. The VIX has some predictive

ability in the medium-term, while the Kelly–Jiang tail-risk measure starts showing some

predictive power after 6 months. The best predictor appears to be the BTX measure from

Bollerslev, Todorov, and Xu (2015). From the third month on, we see a statistically signifi-

cant relation between future CRSP returns and the option-based tail-risk measure. We also

observe a growing pattern in the coefficients. This is also apparent in our Hellinger tail-risk

measure, which could be due to the period starting in 1996, because this pattern is also

present for the regressions with the VIX and not for the other horizons.

We continue our predictive exercise in Table 8, by adding many of the predictive candi-

dates used in Welch and Goyal (2008) to the previous regression with our Hellinger tail-

risk measure and lagged returns for the long sample (1962–2012), that is

r½t;tþk� ¼ aþ bTRt þ cr½t�k;t� þ dGWt þ ut. In Panel A, of the seven predictors considered

(book-to-market, dividend payout, earnings price ratio, dividend price ratio, dividend yield,

Figure 6. This figure plots two series of tail risk. The blue line corresponds to the tail risk computed

from the five principal components extracted monthly from the whole cross-section of CRSP securities

available at each time period with the estimated Hellinger risk-neutral density (c ¼ �0:5). The red line

features the tail-risk measure proposed by Kelly and Jiang (2013) based on the Hill estimator com-

puted from the whole cross-section of CRSP securities available at each time period, based on the Hill

formula featured in Equation (19). The sample period ranges from January 1996 to December 2012.

18 Since results with the S&P500 were very similar, we omitted them for space considerations.

356 Journal of Financial Econometrics

Downloaded from https://academic.oup.com/jfec/article-abstract/15/3/333/3072611by Universite de Montreal useron 19 March 2018

stock variance, default yield spread), only the dividend price ratio and the dividend yield

appear to be significantly related to future stock market returns for all horizons, as it has

been often established. However, it does not take away the predictive power of tail risk for

the 3- to 8-month horizons. The results for the default yield spread are particularly interest-

ing. It shows some significant predictive ability for the 6- to 12-month horizons, but it takes

away the predictive significant relation between tail risk and future market returns. This

Table 7. Predicting market returns

Panel A: Hellinger tail risk

Forecast TR t TR t TR t TR t TR t

1 0.10 (2.90) 0.10 (1.36) 0.10 (2.43) 0.03 (0.52) 0.08 (1.81)

2 0.10 (2.28) 0.11 (1.35) 0.12 (2.45) 0.08 (1.06) 0.10 (1.77)

3 0.10 (2.51) 0.17 (1.97) 0.15 (3.09) 0.12 (1.53) 0.13 (2.35)

4 0.08 (1.90) 0.21 (2.21) 0.15 (2.93) 0.15 (1.78) 0.12 (2.06)

5 0.06 (1.50) 0.23 (2.20) 0.14 (2.62) 0.17 (1.94) 0.11 (1.80)

6 0.03 (0.59) 0.21 (1.69) 0.12 (1.97) 0.15 (1.44) 0.08 (1.21)

7 0.03 (0.57) 0.22 (1.71) 0.12 (1.97) 0.17 (1.52) 0.09 (1.27)

8 0.01 (0.21) 0.21 (1.47) 0.11 (1.64) 0.14 (1.15) 0.08 (1.05)

9 �0.02 (�0.33) 0.18 (1.14) 0.09 (1.20) 0.11 (0.81) 0.06 (0.67)

10 �0.03 (�0.49) 0.20 (1.21) 0.09 (1.08) 0.12 (0.87) 0.06 (0.63)

11 �0.03 (�0.48) 0.20 (1.24) 0.09 (1.03) 0.12 (0.87) 0.05 (0.58)

12 �0.02 (�0.42) 0.19 (1.24) 0.08 (1.00) 0.11 (0.81) 0.05 (0.55)

Panel B: Other tail-risk measures

Forecast Bloom t BTX t KJ t VIX t CATFIN t

1 0.08 (1.98) 0.11 (1.13) 0.05 (1.28) 0.07 (1.00) 0.04 (0.73)

2 0.11 (2.11) 0.15 (1.21) 0.09 (1.67) 0.11 (1.15) 0.04 (0.58)

3 0.10 (1.74) 0.19 (1.95) 0.10 (1.65) 0.15 (1.63) 0.09 (1.08)

4 0.09 (1.37) 0.26 (3.33) 0.12 (1.77) 0.17 (2.01) 0.07 (0.84)

5 0.08 (1.15) 0.30 (3.61) 0.15 (1.89) 0.21 (2.45) 0.08 (0.98)

6 0.08 (0.96) 0.29 (2.87) 0.17 (2.04) 0.22 (2.52) 0.06 (0.75)

7 0.07 (0.77) 0.31 (2.48) 0.19 (2.13) 0.23 (2.31) 0.06 (0.90)

8 0.04 (0.41) 0.30 (2.10) 0.21 (2.22) 0.22 (2.04) 0.06 (0.85)

9 0.03 (0.32) 0.30 (2.11) 0.22 (2.27) 0.20 (1.78) 0.05 (0.65)

10 0.02 (0.20) 0.30 (2.05) 0.23 (2.32) 0.20 (1.67) 0.06 (0.79)

11 0.01 (0.08) 0.32 (2.19) 0.25 (2.49) 0.20 (1.58) 0.06 (0.76)

12 0.01 (0.05) 0.34 (2.29) 0.27 (2.69) 0.21 (1.53) 0.07 (0.88)

Notes: Returns of the CRSP value-weighted market index returns are predicted with the following regressions:

r½t;tþk� ¼ aþ bTRt�1 þ cr½t�k;t� þ ut. We consider the following financial “tail-risk” measures (sample in paren-

thesis) as explanatory variables: Hellinger Tail Risk; Bloom (2009) volatility (07/1962–06/2008); Kelly and

Jiang (2014) tail-risk measure (01/1963–12/2010); Allen, Bale, and Tang (2012) systemic risk measure (01/

1973–12/2012); the VIX index (01/1990–04/2014); and Bollerslev, Todorov, and Xu (2015) tail-risk measure

(01/1996–08/2013). In Panel A, we run the same regression for the Hellinger tail-risk measure for various sam-

ples corresponding to the respective measures considered in Panel B. t-Statistics are calculated using Newey–

West variance matrix with 2–24 lags for 1–12 forecasting horizon.

Almeida et al. j Nonparametric Tail Risk and the Macroeconomy 357

Downloaded from https://academic.oup.com/jfec/article-abstract/15/3/333/3072611by Universite de Montreal useron 19 March 2018

Tab

le8.

We

lsh

an

dG

oy

al(2

00

8)

reg

ress

ion

s

Panel

A:H

ellinger

tail

risk

Fore

cast

TR

tT

Rt

TR

tT

Rt

TR

tT

Rt

TR

t

(BM

)(D

P)

(EPR

)(D

PR

)(D

Y)

(SV

)(D

YS)

10.0

3(0

.58)

0.0

2(0

.42)

0.0

4(0

.77)

0.0

3(0

.59)

0.0

3(0

.59)

0.1

2(2

.19)

0.0

1(0

.18)

20.0

9(1

.62)

0.0

8(1

.36)

0.1

1(1

.96)

0.0

9(1

.61)

0.0

9(1

.63)

0.1

6(3

.21)

0.0

6(1

.14)

30.1

4(2

.62)

0.1

1(2

.17)

0.1

5(2

.96)

0.1

4(2

.65)

0.1

4(2

.70)

0.1

9(3

.19)

0.1

0(1

.92)

40.1

4(2

.59)

0.1

1(2

.01)

0.1

5(2

.86)

0.1

4(2

.60)

0.1

4(2

.68)

0.1

7(2

.82)

0.0

9(1

.67)

50.1

2(2

.55)

0.0

9(1

.82)

0.1

3(2

.82)

0.1

2(2

.59)

0.1

3(2

.73)

0.1

4(2

.31)

0.0

6(1

.26)

60.1

1(2

.42)

0.0

8(1

.61)

0.1

2(2

.65)

0.1

1(2

.50)

0.1

2(2

.65)

0.1

2(2

.00)

0.0

5(1

.04)

70.1

0(2

.01)

0.0

7(1

.38)

0.1

1(2

.25)

0.1

0(2

.13)

0.1

1(2

.28)

0.0

9(1

.44)

0.0

4(0

.85)

80.1

0(1

.98)

0.0

4(1

.32)

0.1

1(2

.21)

0.1

0(2

.10)

0.1

1(2

.25)

0.0

8(1

.31)

0.0

4(0

.83)

90.1

0(1

.74)

0.0

6(1

.20)

0.1

1(2

.00)

0.1

0(1

.86)

0.1

1(2

.00)

0.0

7(1

.11)

0.0

3(0

.70)

10

0.0

8(1

.35)

0.0

5(0

.82)

0.1

0(1

.60)

0.0

9(1

.46)

0.0

9(1

.60)

0.0

5(0

.78)

0.0

2(0

.33)

11

0.0

8(1

.18)

0.0

4(0

.66)

0.0

9(1

.42)

0.0

8(1

.28)

0.0

9(1

.42)

0.0

4(0

.58)

0.0

1(0

.18)

12

0.0

8(1

.13)

0.0

4(0

.58)

0.0

9(1

.39)

0.0

8(1

.22)

0.0

9(1

.39)

0.0

3(0

.48)

0.0

1(0

.11)

Fore

cast

BM

tD

Pt

EPR

tD

PR

tD

Yt

SV

tD

YS

t

10.0

6(1

.30)

0.0

2(0

.35)

0.0

7(1

.39)

0.0

9(2

.16)

0.0

9(2

.17)

�0.1

8(�

3.7

9)

0.0

6(1

.04)

20.0

8(1

.45)

0.0

3(0

.45)

0.1

0(1

.40)

0.1

3(2

.40)

0.1

3(2

.41)

�0.1

5(�

2.2

2)

0.0

7(0

.90)

30.1

1(1

.61)

0.0

4(0

.69)

0.1

2(1

.44)

0.1

6(2

.59)

0.1

6(2

.58)

�0.1

3(�

1.4

5)

0.0

9(1

.03)

40.1

3(1

.59)

0.0

6(0

.92)

0.1

3(1

.32)

0.1

9(2

.54)

0.1

9(2

.55)

�0.1

0(�

0.9

5)

0.1

3(1

.42)

50.1

4(1

.56)

0.0

8(1

.10)

0.1

4(1

.24)

0.2

1(2

.45)

0.2

1(2

.51)

�0.0

8(�

0.9

2)

0.1

6(1

.75)

60.1

6(1

.55)

0.0

9(1

.16)

0.1

5(1

.25)

0.2

2(2

.39)

0.2

3(2

.44)

�0.0

5(�

0.7

4)

0.1

7(1

.80)

70.1

7(1

.56)

0.0

9(1

.22)

0.1

6(1

.30)

0.2

4(2

.37)

0.2

5(2

.40)

�0.0

2(�

0.3

6)

0.1

8(1

.89)

80.1

8(1

.56)

0.0

9(1

.21)

0.1

7(1

.39)

0.2

6(2

.35)

0.2

6(2

.38)

0.0

0(�

0.0

5)

0.1

8(1

.89)

90.1

9(1

.56)

0.0

9(1

.20)

0.1

8(1

.50)

0.2

7(2

.36)

0.2

7(2

.38)

0.0

1(0

.19)

0.1

9(1

.97)

10

0.1

9(1

.56)

0.1

0(1

.19)

0.1

9(1

.58)

0.2

8(2

.37)

0.2

8(2

.39)

0.0

1(0

.25)

0.2

0(2

.10)

11

0.2

0(1

.56)

0.1

0(1

.09)

0.2

1(1

.70)

0.2

9(2

.38)

0.2

9(2

.40)

0.0

2(0

.49)

0.2

1(2

.12)

12

0.2

0(1

.53)

0.1

0(1

.03)

0.2

2(1

.79)

0.3

0(2

.38)

0.3

1(2

.42)

0.0

3(0

.68)

0.2

0(2

.05)

(conti

nued

)

358 Journal of Financial Econometrics

Downloaded from https://academic.oup.com/jfec/article-abstract/15/3/333/3072611by Universite de Montreal useron 19 March 2018

Tab

le8.

Co

nti

nu

ed

Panel

B:C

ontr

olvari

able

s

Fore

cast

TR

tT

Rt

TR

tT

Rt

TR

tT

Rt

TR

t

(DR

S)

(Infl

.)(L

TY

)(L

TR

R)

(TS)

(TB

R)

(NE

E)

10.0

2(0

.37)

0.0

2(0

.34)

0.0

3(0

.52)

0.0

2(0

.30)

0.0

2(0

.34)

0.0

3(0

.50)

0.0

2(0

.44)

20.0

8(1

.37)

0.0

8(1

.51)

0.0

9(1

.52)

0.0

8(1

.32)

0.0

7(1

.27)

0.0

9(1

.54)

0.0

8(1

.50)

30.1

2(2

.28)

0.1

2(2

.38)

0.1

3(2

.45)

0.1

2(2

.25)

0.1

1(2

.17)

0.1

3(2

.48)

0.1

2(2

.50)

40.1

2(2

.24)

0.1

2(2

.24)

0.1

3(2

.41)

0.1

2(2

.21)

0.1

1(2

.13)

0.1

3(2

.44)

0.1

2(2

.50)

50.1

0(2

.14)

0.1

0(2

.04)

0.1

1(2

.30)

0.1

0(2

.07)

0.0

9(1

.99)

0.1

1(2

.32)

0.1

0(2

.19)

60.0

9(1

.99)

0.0

8(1

.86)

0.1

0(2

.11)

0.0

9(1

.90)

0.0

8(1

.76)

0.1

0(2

.11)

0.0

9(1

.83)

70.0

8(1

.65)

0.0

7(1

.51)

0.0

9(1

.73)

0.0

8(1

.54)

0.0

7(1

.42)

0.0

9(1

.72)

0.0

8(1

.47)

80.0

8(1

.64)

0.0

7(1

.44)

0.0

9(1

.69)

0.0

8(1

.54)

0.0

7(1

.34)

0.0

9(1

.67)

0.0

8(1

.42)

90.0

8(1

.43)

0.0

7(1

.27)

0.0

9(1

.45)

0.0

7(1

.32)

0.0

6(1

.16)

0.0

9(1

.46)

0.0

7(1

.22)

10

0.0

6(1

.04)

0.0

5(0

.89)

0.0

7(1

.09)

0.0

6(0

.94)

0.0

4(0

.79)

0.0

7(1

.10)

0.0

6(0

.89)

11

0.0

6(0

.85)

0.0

5(0

.73)

0.0

7(0

.93)

0.0

5(0

.75)

0.0

4(0

.61)

0.0

7(0

.93)

0.0

5(0

.72)

12

0.0

5(0

.78)

0.0

4(0

.63)

0.0

6(0

.87)

0.0

4(0

.67)

0.0

3(0

.52)

0.0

6(0

.88)

0.0

5(0

.65)

(conti

nued

)

Almeida et al. j Nonparametric Tail Risk and the Macroeconomy 359

Downloaded from https://academic.oup.com/jfec/article-abstract/15/3/333/3072611by Universite de Montreal useron 19 March 2018

Tab

le8.

Co

nti

nu

ed

Fore

cast

DR

St

Infl

.t

LT

Yt

LT

RR

tT

St

TB

Rt

NE

Et

1�

0.0

5(�

1.1

8)

�0.0

4(�

0.8

8)

0.0

3(0

.81)

0.0

9(2

.10)

0.0

5(1

.27)

0.0

1(0

.28)

�0.0

1(�

0.2

5)

2�

0.0

4(�

0.7

7)

0.0

0(�

0.0

5)

0.0

6(1

.06)

0.0

9(2

.00)

0.0

6(1

.04)

0.0

3(0

.56)

�0.0

1(�

0.0

8)

3�

0.0

1(�

0.2

8)

�0.0

1(�

0.1

1)

0.0

8(1

.15)

0.0

7(1

.49)

0.0

5(0

.82)

0.0

4(0

.64)

0.0

1(0

.07)

4�

0.0

2(�

0.3

6)

�0.0

2(�

0.3

1)

0.0

8(1

.11)

0.0

9(1

.84)

0.0

6(0

.84)

0.0

5(0

.59)

0.0

1(0

.08)

5�

0.0

4(�

0.7

6)

�0.0

4(�

0.6

3)

0.1

0(1

.14)

0.1

3(3

.08)

0.0

8(0

.97)

0.0

5(0

.61)

0.0

0(�

0.0

1)

6�

0.0

4(�

0.6

7)

�0.0

5(�

0.6

6)

0.1

2(1

.28)

0.1

4(3

.22)

0.0

9(1

.04)

0.0

7(0

.69)

�0.0

1(�

0.0

7)

7�

0.0

3(�

0.6

2)

�0.0

6(�

0.8

4)

0.1

4(1

.46)

0.1

5(3

.41)

0.1

0(1

.12)

0.0

8(0

.78)

�0.0

1(�

0.0

8)

8�

0.0

1(�

0.1

8)

�0.0

7(�

0.9

6)

0.1

6(1

.68)

0.1

4(3

.33)

0.1

2(1

.19)

0.0

9(0

.87)

�0.0

1(�

0.0

8)

90.0

1(0

.15)

�0.0

7(�

0.9

3)

0.1

8(1

.87)

0.1

3(3

.22)

0.1

3(1

.33)

0.1

0(0

.94)

�0.0

2(�

0.1

2)

10

0.0

3(0

.57)

�0.0

7(�

0.9

4)

0.2

0(2

.05)

0.1

1(2

.88)

0.1

4(1

.46)

0.1

0(1

.01)

�0.0

2(�

0.1

4)

11

0.0

4(0

.66)

�0.0

7(�

0.8

9)

0.2

1(2

.26)

0.1

1(2

.99)

0.1

5(1

.54)

0.1

1(1

.16)

�0.0

2(�

0.1

3)

12

0.0

4(0

.71)

�0.0

8(�

1.0

1)

0.2

3(2

.49)

0.1

2(3

.55)

0.1

6(1

.63)

0.1

2(1

.32)

�0.0

3(�

0.1

4)

Note

s:R

eturn

sof

the

CR

SP

valu

e-w

eighte

dm

ark

etin

dex

retu

rns

are

pre

dic

ted

wit

hth

efo

llow

ing

regre

ssio

ns:

r ½t;

tþk�¼

aþ

bTR

tþ

cr½t�

k;t�þ

dGW

tþ

ut,

wher

eT

Rin

dic

ate

sth

e

Hel

linger

Tail

Ris

k.A

ddit

ionalco

ntr

ols

(GW

)in

dic

ate

the

Goyaland

Wel

ch(2

008)

surv

eyed

pre

dic

tors

:book-t

o-m

ark

et(B

M),

div

iden

dpayout

(DP),

earn

ings

pri

cera

tio

(EPR

),div

-

iden

dpri

cera

tio

(DPR

),div

iden

dyie

ld(D

Y),

stock

vari

ance

(SV

),def

ault

yie

ldsp

read

(DY

S),

def

ault

retu

rnsp

read

(DR

S),

inflati

on

(Infl.)

,lo

ng-t

erm

yie

ld(L

TY

),lo

ng-t

erm

rate

of

retu

rns

(LT

RR

),te

rmsp

read

(TS),

trea

sury

bill

rate

(TB

R),

and

net

equit

yex

pansi

on

(NE

E).

Inboth

panel

sA

and

B,w

epre

sent

the

esti

mate

dco

effici

ents

for

the

Hel

linger

Tail

Ris

k

inth

eupper

part

and

the

esti

mate

dco

effici

ent

for

the

addit

ional

contr

ol

vari

able

sin

the

low

erpart

.t-

Sta

tist

ics

are

calc

ula

ted

usi

ng

New

ey–W

est

vari

ance

matr

ixw

ith

2–24

lags

for

1–12

fore

cast

ing

hori

zon.Sam

ple

:Ju

ly1962

toD

ecem

ber

2012.

360 Journal of Financial Econometrics

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reinforces the interpretation of our measure as tail risk since the default spread ought to

widen in periods when tail risk is prevalent. It is therefore hard to capture each effect

independently.

In Panel B of Table 8, we consider another set of seven predictors used in Welch and

Goyal (2008): default return spread, inflation, long-term yield, long-term rate of returns,

term spread, treasury bill rate, and net equity expansion. Only the long-term rate of bond

returns shows a strong and statistically significant predictive power, but tails risk remains

as a significant predictor 3-to-6 months ahead.

Overall, we can conclude that tail risk has a significant predictive power in the short run

even after controlling for the main predictors of future stock market returns.

3.2. Economic Predictability

We now explore the relationship of our tail-risk measure with the real economy. Recently,

in the aftermath of the great recession, several papers put forward facts and patterns about

economic uncertainty, its fluctuations during business cycles, and its empirical relation with

micro and macro growth dispersion measures. While measures of volatility may capture

uncertainty, it is often the case that big negative shocks play an important role in increasing

economic uncertainty. This asymmetry occurs because good news seem to build up more

gradually over a period of time. Therefore, tail-risk measures may help in measuring better

the effects of economic uncertainty on macroeconomic variables such as output growth and

employment. Both Bloom (2009) and Bali, Brown, and Caglayan (2014) propose measures

of macroeconomic uncertainty based on volatility of macroeconomic variables. We have

seen that our measure of tail risk has a correlation close to 0.5 with these two uncertainty

indicators. We have also characterized the jumps in tail risk in Figure 3 and they often cor-

respond to jumps in uncertainty identified by Bloom (2009).19 In what follows we will fol-

low the vector auto-regressive impulse response function approach of Bloom (2009) to

distinguish between the effects of volatility and tail risk on macroeconomic aggregate

variables.

3.2.1 Impulse response functions to tail-risk shocks

To assess whether uncertainty shocks influenced firm-level decisions, Bloom (2009) consid-

ered several variables in his VAR approach: stock market returns, volatility(either market

variance or an indicator capturing the biggest volatility events), the Federal Funds rate,

wages, inflation, hours, employment, and production in manufacturing, in this order. A

main conclusion was that a rise in economic uncertainty reduces industrial production and

employment in the short run. We argue that some of the uncertainty effect captured by

Bloom’s study might actually be coming from tail risk, in the same line as Kelly and Jiang

(2014).20 The main theoretical insight behind our argument comes from a firm’s

19 Bloom (2009) identifies 17 uncertainty dates defined as events associated with stock market vola-

tility in excess of 1.65 standard deviations above its trend.20 These authors use the tail-risk measure that we described earlier applied to the manufacturing

firms in the sample. It is therefore a very specific measure that we want to compare to our

general-purpose measure based on principal components. It is also instructive to see how our

measure performs given the two very different properties of the two measures as illustrated in

Section 2.3.

Almeida et al. j Nonparametric Tail Risk and the Macroeconomy 361

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perspective on investment decisions in the presence of uncertainty. If firms look at their

investment choices as real options, an increase in uncertainty may defer her decision to

invest, to produce, or to hire. Therefore, a rise in volatility or tail risk may affect aggregate

production or employment as well as investment.21

Therefore, we expand Bloom (2009)‘s vector auto-regressive approach by adding our

measure of tail risk to the set of variables considered.22 In terms of order we include it just

after volatility. In Figure 7, we plot the impulse response functions on both employment

and industrial production to a one standard deviation shock in the monthly estimated tail-

risk measure.23 The gray area represents the 95% confidence interval for the response func-

tion while the solid line represents the response itself. In Figure 8, we plot the same response

functions for a one standard deviation shock to volatility measured by stock market

variance.24 For the shocks to tail risk, the effect is less abrupt but it seems to be followed by

Figure 7. This figure presents the impulse response functions for Employment and Industrial

Production following a shock in the Hellinger Tail Risk. Results follow from a vector auto-regressive

approach similar to Bloom (2009) including an additional Tail-Risk series (still controlling for the mar-

ket variance).

21 Kellogg (2014) and Jurado, Ludvigson, and Ng (2015) explore in more depth this particular

channel.

22 To be precise, market returns measured by S&P 500 returns lagged, wages, hours, industrial pro-

duction, and employment are in logs, inflation is measured as the difference in log CPI. The varia-

bles are in the order indicated in the previous paragraph.

23 The only difference between our approach and Bloom’s one is that we perform the impulse

response analysis with an additional tail-risk factor.

24 Bloom (2009) uses realized variance before the availability of VXO and VXO afterwards.

362 Journal of Financial Econometrics

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a more positive recovery compared with a volatility shock where after 10 months the strong

negative shock has been absorbed without any ensuing significant positive effect. Although

the evidence is not strong, it seems that a tail-risk shock has a recovery period after the ini-

tial negative shock. In Figures 9 and 10, we plot the same respective impulse response func-

tions but this time with Bloom (2009) indicator of big volatility events. Although the

shapes of the reaction functions are similar, the confidence intervals are now more telling

about the effects of shocks to volatility and tail risk. The responses to tail risk are of greater

magnitude, distinctively negative in the short run up to 10–15 months, and then becoming

significantly positive at an horizon of 30–35 months. For shocks to the volatility indicator,

small initial negative effects die after 5 months and never pick up significantly afterwards

for employment. However, for industrial production, after a small contraction in the first 3

months, there is a small rebound at an horizon of 10–15 months. These results tend to

show that it is difficult to disentangle uncertainty created by big events from the uncertainty

associated with more volatile environments.

3.2.2 Business cycles

Bloom (2009) states that almost every macroeconomic indicator of uncertainty appears to

be countercyclical. In Figure 3, we plot our tail risk, in blue, and highlight with shades of

gray the NBER recession periods. To give a better idea of the relation between tail risk and

recessions we also plot in red the same tail-risk measure after passing the series through a

Hodrick–Prescott filter to isolate the cyclical component. This makes clear that usually tail

risk is higher whenever the economy is in a recession. Therefore, a natural question to ask

Figure 8. This figure presents the impulse response functions for Employment and Industrial

Production following a shock in the market variance. Results follow from a vector auto-regressive

approach similar to Bloom (2009) including an additional Tail-Risk series.

Almeida et al. j Nonparametric Tail Risk and the Macroeconomy 363

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is whether our measure of tail risk has some predictive power over economic downturns.

Our approach is comparable to Allen, Bali, and Tang (2012) who consider measures of sys-

temic risk for the financial sector as predictors for economic downturns. While they focus

their argument on the special intermediation role of the financial industry, our measure is

economy-wide since it is based on portfolios of all firms making up the stock market.

Therefore, in the same logic evoked in the previous section our tail risk should anticipate

movements in the real activity or business cycle indicators.

We collect a large set of macroeconomic indicators and assess their predictability by our

tail-risk index at monthly horizons between 1 and 12 months. Overall, we have eight indi-

ces that are available for different samples: 1) ADS, the Aruoba, Diebold and Scotti macro-

economic activity indicator (02/1960–04/2014); 2) KCFED, the Kansas City FED

macroeconomic indicator index (01/1990–04/2014); the NBER recession indicator (a reces-

sion period dummy, 07/1926–04/2014); CFNAI, the Chicago FED National Activity Index

(02/1967–04/2014); the financial and macroeconomic indexes of Jurando, Ludvigson, and

Ng (2015) (07/1960–04/2014); the St. Louis FED Financial Stress Index (01/1994–04/

2014); and EPU, the Economic Policy Uncertainty Index of Backer, Bloom, and Davis

(2015) (01/1985–04/2014). We run the following regression:

Itþi ¼ b0 þ bTRt þX11

k¼0

ciIt�k þ �t; (20)

Figure 9. This figure presents the impulse response functions for Employment and Industrial Production fol-

lowing a shock in the Hellinger Tail Risk. Results follow from a vector auto-regressive approach similar to

Bloom (2009) including an additional Tail-Risk series (still controlling for the big volatility events indicator).

364 Journal of Financial Econometrics

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where I stands for the activity index we consider and i indexes the forecasting horizon

(i ¼ 1; . . . ;12). Note that since the NBER recession indicator is a binary variable we esti-

mate a Probit model. Regression results are reported in Table 9. We observe that for all

indicators except the KCFED there is a significant predictability at short horizons up to 3–4

months ahead. For the NBER and the financial uncertainty indicator the predictive power

is strong and extends to all horizons up to 12 months ahead. The signs are of course nega-

tive to capture the negative effect of current tail-risk increases on future activity, except of

course for NBER recessions and EPU which are downturn and uncertainty measures,

respectively.

To conclude this section we discuss the inter-temporal relationship between our tail-risk

measure and four other tail-risk measures over different time periods,25 namely the Bali,

Brown, and Caglayan (2014) macroeconomic uncertainty index (01/1994–12/2013); the

BTX tail-risk index (01/19963–08/2013) in Bollerslev, Todorov, and Xu (2015); the Kelly

and Jiang (2014) tail-risk measure (01/1963–12/2010); and finally the Allen, Bali, and

Tang (2012) systemic risk measure (01/1973–12/2012) called CATFIN. We use the same

prediction regression framework as in Equation (20). In Table 10, it can be seen that we

predict well the Macro and KJ indicators at all horizons. For the BTX index, the predict-

ability is limited to the very short horizons, while for CATFIN we see some predictability in

Figure 10. This figure presents the impulse response functions for Employment and Industrial

Production following a shock in the big volatility event indicator. Results follow from a vector auto-

regressive approach similar to Bloom (2009) including an additional Tail-Risk series.

25 These time periods are determined by the currently available series on the web sites of the

researchers.

Almeida et al. j Nonparametric Tail Risk and the Macroeconomy 365

Downloaded from https://academic.oup.com/jfec/article-abstract/15/3/333/3072611by Universite de Montreal useron 19 March 2018

Tab

le9.

Ma

cro

eco

no

mic

act

ivit

yp

red

icti

on

reg

ress

ion

s

Panel

A

Fore

cast

AD

St

R2

KC

FE

Dt

R2

NB

ER

tR

2C

FN

AI

tR

2

1�

0.0

5(�

2.7

0)

0.8

40.0

1(�

0.2

7)

0.9

20.6

9�

3.1

70.7

8�

0.1

0(�

2.8

0)

0.5

3

2�

0.0

9(�

3.0

9)

0.5

2�

0.0

3(�

0.5

5)

0.8

10.5

2�

2.5

00.6

2�

0.1

2(�

3.3

7)

0.4

6

3�

0.0

8(�

2.4

0)

0.3

9�

0.0

9(�

1.0

5)

0.7

40.5

8�

3.2

70.5

0�

0.1

1(�

2.9

7)

0.3

6

4�

0.0

5(�

1.5

7)

0.3

0�

0.1

6(�

1.3

9)

0.6

50.5

4�

3.4

50.4

0�

0.0

8(�

1.8

9)

0.2

7

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0.0

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0.1

7(�

1.4

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0.5

60.4

5�

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40.3

0�

0.0

6(�

1.6

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6�

0.0

3(�

0.7

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0.2

0(�

1.4

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90.4

3�

3.2

70.2

2�

0.0

7(�

1.8

9)

0.1

6

7�

0.0

2(�

0.4

5)

0.1

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0.2

1(�

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3.6

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5�

0.0

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0.0

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0�

0.0

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(conti

nued

)

366 Journal of Financial Econometrics

Downloaded from https://academic.oup.com/jfec/article-abstract/15/3/333/3072611by Universite de Montreal useron 19 March 2018

Tab

le9.

Co

nti

nu

ed

Panel

B

Fore

cast

Fin

.U

nce

rt.

tR

2M

acr

oU

nce

rt.

tR

2St.

Louis

tR

2E

PU

tR

2

1�

0.2

2(�

2.4

4)

0.3

2�

0.1

0(�

2.3

5)

0.4

2�

0.1

7(�

1.8

1)

0.1

50.0

8(4

.18)

0.7

5

2�

0.2

6(�

4.1

7)

0.1

4�

0.1

6(�

2.5

9)

0.1

4�

0.2

5(�

2.2

1)

0.1

00.0

9(3

.80)

0.6

1

3�

0.2

0(�

4.4

7)

0.0

7�

0.1

4(�

2.1

7)

0.0

7�

0.2

4(�

2.5

1)

0.0

80.1

1(3

.15)

0.5

6

4�

0.1

6(�

3.4

0)

0.0

5�

0.1

7(�

2.1

6)

0.0

4�

0.2

5(�

2.4

7)

0.0

90.0

3(0

.82)

0.5

2

5�

0.1

4(�

2.6

6)

0.0

4�

0.1

3(�

1.6

6)

0.0

3�

0.1

7(�

1.6

3)

0.0

60.0

5(1

.05)

0.4

8

6�

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2(�

2.6

3)

0.0

3�

0.1

2(�

1.8

3)

0.0

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1.7

9)

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4(0

.90)

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4

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.24)

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3

8�

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6(�

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8)

0.0

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1(�

1.9

0)

0.0

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0.0

4(�

0.5

5)

0.0

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.05)

0.4

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9�

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1(�

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1(�

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0.0

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1.0

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0.0

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.86)

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1

10

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0(�

3.0

5)

0.0

2�

0.0

9(�

1.9

0)

0.0

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4(�

2.2

3)

0.0

50.0

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.05)

0.4

0

11

�0.1

5(�

3.0

6)

0.0

3�

0.0

8(�

2.1

8)

0.0

3�

0.0

2(�

0.3

5)

0.0

30.1

1(1

.75)

0.3

8

12

�0.0

7(�

1.8

1)

0.0

2�

0.0

5(�

1.5

2)

0.0

30.0

2(0

.59)

0.0

40.1

3(2

.43)

0.3

4

Note

s:Pre

dic

tive

regre

ssio

ns

are

run

for

avari

ety

of

macr

oec

onom

icin

dic

ato

rs(e

stim

ati

ng

sam

ple

inpare

nth

eses

).A

DS

repre

sents

the

Aru

oba,

Die

bold

,and

Sco

tti

macr

oec

onom

ic

act

ivit

yin

dic

ato

r(0

2/1

960–04/2

014);

KC

FE

Dre

pre

sents

the

Kansa

sC

ity

FE

Dm

acr

oec

onom

icin

dic

ato

rin

dex

(01/1

990–04/2

014);

NB

ER

repre

sents

are

cess

ion

per

iod

dum

my

(07/

1926–04/2

01

4);

CFN

AI

repre

sents

the

Chic

ago

FE

DN

ati

onal

Act

ivit

yIn

dex

(02/1

967–04

/2014);

Fin

and

Macr

oU

nce

rt.

indic

ate

sJu

rando,

Ludvig

son,

and

Ng

(2015)

financi

al

and

macr

oec

onom

icunce

rtain

tyin

dex

,re

spec

tivel

y(0

7/1

960–04/2

014);

St.

Louis

stands

for

the

St.

Louis

FE

DFin

anci

al

Str

ess

Index

(01/1

994–04/2

014);

and

EPU

repre

sents

the

Eco

nom

icPolicy

Unce

rtain

tyIn

dex

of

Back

er,

Blo

om

,and

Davis

(2015)

(01/1

985–04/2

014).

All

regre

ssio

ns

contr

ol

for

12

lags

of

the

endogen

ous

vari

able

.A

llt-

stati

stic

sare

com

-

pute

dw

ith

a24-l

ag

New

ey–W

est

corr

ecti

on.For

the

NB

ER

vari

able

,w

ere

port

the

Pro

bit

regre

ssio

nre

sult

s.

Almeida et al. j Nonparametric Tail Risk and the Macroeconomy 367

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the first 6 months. We also note that the relationship with Bollerslev, Todorov, and Xu

(2015) is almost monotonically decreasing, but that it is the contrary for the Kelly and

Jiang (2014) measure. This is consistent with the fact that the KJ tail risk is a slow moving,

very persistent measure. On the contrary, the BTX measure of Bollerslev, Todorov, and Xu

(2015), like ours, is very volatile and quickly mean reverts.

4 Robustness

To build our measure of tail risk, we have made several choices regarding the parameter of

the Cressie-Read discrepancy family (c ¼ �0:5, so-called Hellinger discrepancy), the number

of principal components of the 25 size and book-to-market portfolios, the number of days

over which the excess ES is computed (30 days), and the threshold of the shortfall set at 10%.

Therefore, we will first verify whether our results are sensitive to these choices. Second, we

argued that there were good reasons for choosing an ES measure over a VaR measure.

However, other papers have built tail or downside risk VaR-based measures and shown that

they were superior to other risk measures (see in particular Bali, Demirtas, and Levy, 2009).

We will then reproduce our results for VaR measures and compare them to our original

results with ES. Finally, our benchmark measure has been obtained from a set of five princi-

pal components of the 25 size and book-to-market portfolios. We already checked that the

sorting portfolio results survived when we extracted the principal components from the

whole cross-section of individual stocks. Another important question concerns the choice of

portfolios. Will the measure keep similar properties when constructed from industry portfo-

lios, or from the financial sector only, for example? This will be our third line of investigation

for measuring the robustness of our methodology for measuring tail risk.

Table 10. Prediction of other tail-risk measures

Forecast Macro t R2 BTX t R2 KJ t R2 CATFIN t R2

1.00 0.04 (�1.88) 0.96 0.32 (�2.14) 0.48 0.02 (�1.17) 0.88 �0.01 (�0.33) 0.54

2.00 0.11 (�1.80) 0.92 0.18 (�2.94) 0.30 0.03 (�1.33) 0.82 �0.05 (�1.87) 0.43

3.00 0.18 (�2.18) 0.88 0.15 (�1.18) 0.15 0.04 (�2.05) 0.80 �0.07 (�1.62) 0.39

4.00 0.18 (�2.40) 0.83 0.08 (�1.90) 0.11 0.05 (�2.11) 0.77 �0.06 (�1.41) 0.34

5.00 0.18 (�2.83) 0.77 0.14 (�1.34) 0.10 0.05 (�1.82) 0.74 �0.14 (�3.13) 0.34

6.00 0.17 (�3.11) 0.71 0.08 (�1.46) 0.06 0.08 (�2.69) 0.72 �0.12 (�2.06) 0.30

7.00 0.17 (�3.41) 0.65 0.20 (�2.85) 0.08 0.09 (�2.67) 0.69 �0.09 (�1.36) 0.28

8.00 0.15 (�3.15) 0.59 0.09 (�1.14) 0.04 0.08 (�2.45) 0.66 �0.06 (�0.91) 0.27

9.00 0.15 (�2.98) 0.53 0.11 (�1.56) 0.05 0.08 (�2.30) 0.65 �0.04 (�0.75) 0.24

10.00 0.14 (�2.62) 0.46 0.13 (�1.84) 0.05 0.09 (�2.29) 0.62 �0.03 (�0.63) 0.22

11.00 0.14 (�2.77) 0.39 0.08 (�0.99) 0.04 0.13 (�2.88) 0.61 �0.03 (�0.50) 0.20

12.00 0.15 (�2.82) 0.34 0.15 (�1.65) 0.05 0.15 (�3.38) 0.59 �0.04 (�0.85) 0.17

Notes: Predictive regressions are run for a variety of Tail-Risk indexes using the Hellinger Tail Risk as a predic-

tor. KJ refers to Kelly and Jiang (2014) tail-risk index (01/1963–12/2010); BTX refers to Bollerslev, Todorov,

and Xu (2015) tail-risk index (01/19963–08/2013); Macro refers to Bali, Brown, and Caglayan (2014) macro-

economic uncertainty index (01/1994–12/2013); and CAFTIN denotes Allen, Bali, and Tang (2012) systemic

risk measure (01/1973–12/2012). All regressions control for 12 lags of the endogenous variable. Bali, Brown,

and Caglayan (2014) is I(1) so we perform the regression in first differences. All t statistics are computed with

a 24-lag Newey–West correction.

368 Journal of Financial Econometrics

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4.1. Robustness to the Parameters of the Measure

In Table 11, we report the returns of the Long-Short portfolios associated with different

discrepancy parameters (first four columns). We can see that the patterns and magnitudes

of returns are very similar to the benchmark results. The statistical significance of the last

set of control variables, which includes volatility, disappears for 3 out of 4. The discrepancy

closest to ours in the negatives, c ¼ �1, provides therefore the most similar results. Adding

principal components (10 instead of 5) does not improve results, nor increasing nor reduc-

ing the number of days over which we compute the ES. Again, we seem to lose precision in

estimating the risk premiums, especially after controlling for volatility. Finally, setting the

Table 11. Robustness: sorted portfolio

Panel A: One month holding period

c ¼ �3 c ¼ �1 c¼ 0 c¼ 1 10 PC 45 D. 20 D. 5%

Average return �1.55 �1.62 �1.42 �1.37 �1.28 �1.68 �1.29 �1.52

�3.46 �3.61 �3.32 �3.16 �2.86 �3.68 �2.92 �3.34

FF3 �1.17 �1.24 �1.10 �1.05 �0.91 �1.19 �0.79 �0.99

�3.08 �3.35 �3.01 �2.73 �2.37 �3.02 �2.20 �2.61

FF3þMOM �1.22 �1.29 �1.15 �1.07 �0.93 �1.19 �0.86 �1.03

�3.12 �3.36 �3.02 �2.71 �2.38 �2.96 �2.32 �2.63

FF3þMOMþLIQ �1.29 �1.39 �1.21 �1.13 �1.02 �1.29 �0.98 �1.13

�2.82 �3.04 �2.67 �2.41 �2.20 �2.72 �2.18 �2.44

FF3þMOMþLIQþVOL �0.83 �0.96 �0.74 �0.60 �0.30 �0.53 �0.57 �0.68

�1.79 �2.09 �1.62 �1.36 �0.70 �1.23 �1.26 �1.63

Panel B: One year holding period

Average return �14.67 �14.60 �12.27 �12.43 �9.42 �15.57 �11.32 �14.71

�3.06 �3.23 �2.88 �3.02 �2.41 �3.54 �2.51 �3.63

FF3 �3.07 �3.78 �2.17 �2.19 0.56 �3.63 �0.29 �4.18

�1.01 �1.38 �0.82 �0.84 0.22 �1.33 �0.10 �1.55

FF3þMOM �5.50 �5.68 �4.08 �3.30 �1.61 �4.54 �2.42 �4.88

�2.02 �2.29 �1.71 �1.40 �0.68 �1.87 �0.97 �1.94

FF3þMOMþLIQ �6.10 �5.82 �3.76 �3.09 �2.05 �4.82 �2.67 �6.76

�2.24 �2.40 �1.65 �1.29 �0.90 �1.88 �1.18 �2.79

FF3þMOMþLIQþVOL 9.60 10.06 13.37 13.27 8.67 8.63 17.32 8.13

0.80 0.91 1.20 1.16 0.92 0.67 1.38 0.84

Notes: Returns are reported for decile portfolios of CRSP stocks with code 10-11 sorted according to their tail-

risk hedging capacity. The columns indicate differences between the tail-risk measure considered and the base-

line case (c ¼ �0:5, 30 days returns, five principal components, and 10% VaR threshold). PC denotes the num-

ber of principal components used, D the number of days, and 5% the VaR threshold. For each month in our

sample from February 1967 to December 2013, we sort the stocks and track their returns one-month post for-

mation (Panel A) or one-year post formation (Panel B). In the first line we report the average portfolio returns.

In the following lines we report the a0s of regressions where we control for the factors indicated in the first col-

umn. We only report the values for the high-minus-low portfolios. Newey–West t-statistics reported between

parentheses are computed with one lag for monthly results and 12 lags for yearly results.

Almeida et al. j Nonparametric Tail Risk and the Macroeconomy 369

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threshold at 5% instead of 10% does not have much of an effect. We have also computed

the sensitivity of our prediction results to these different values of the parameters.

For both market returns prediction and macroeconomic indicators forecast we arrived

at very similar results for all configurations and all variables. We can then confidently con-

clude that the results are very robust to the parameters used to construct our tail-risk meas-

ure. One could add that they are too robust in the sense that the risk neutralization could

be done indifferently with any gamma value. In this regard it is important to emphasize

that our performance criterion is a simple average where the timing of the prediction is not

taken into account. If we consider that errors are more costly in bad times, then more

important differences will appear between the various discrepancy gamma parameters.

Risk neutralization will matter more.

4.2. Measuring Tail Risk from Industry and Sector Portfolios

We chose to build our benchmark tail-risk measure on the first five principal components

of 25 size and book-to-market portfolios. In this section, we want to illustrate that our

results in terms of risk premia and predictability are not changed significantly if we take

other reference portfolios. We have seen in Section 2.3 that risk premia on the high-minus-

low portfolios were similar when the five principal components were computed from the

whole set of securities available in the cross-section. Here, we choose three sets of portfo-

lios: a set of 10 industry portfolios, an aggregate portfolio of the financial sector, and an

aggregate portfolio of the real sector. Results are reported in Table 12. Apart from the sort-

ing portfolios columns where alphas are reported for the high-minus-low portfolio we also

feature market returns predictions, in a long sample [similar to Kelly and Jiang (2014) start-

ing in the 60s] and a short one [as in Bollerslev, Todorov, and Xu (2015) starting in the

90s]. For the long sample we also show market prediction results controlling for dividend

yield and stock variance as predictors.

In the first panel featuring 10 industry portfolios, risk premiums are remarkably similar

in terms of magnitudes and statistical significance to our benchmark estimates in Table 4.

Note that the sign of the one-year holding period when controlling for all factors including

volatility is now negative, although not significant. For the prediction of market returns,

the statistical significance patterns are very similar, where the short-term prevails, but the

regression coefficients are lower in general.

In the financial sector panel, risk premia are generally higher in absolute value, espe-

cially for the long holding period, but the statistical significance remains quite comparable.

The predictive power of market returns disappears in the short sample (after the 90s) which

may be explained by the turmoil in financial markets during this period. We also lose some

significance and impact when controlling for stock variance. This is to be expected since

tail-risk measured from financial returns is harder to disentangle from the high stock var-

iance episodes. This interpretation is supported by the evidence in the last panel when we

consider tail risk measured from the real sector alone.

The risk premia are again very close to the benchmark values in Table 4 with a slightly

higher level of precision in the estimates. The predictive power is also estimated with much

more precision, especially when controlling for other predictors.

We did not include results on economic predictability because they were very similar to

the benchmark results with the size and book-to-market portfolios.

370 Journal of Financial Econometrics

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Tab

le12.

Ro

bu

stn

ess

:in

du

stry

po

rtfo

lio

s

10

indust

ryport

folios

Ret

urn

pre

dic

tion

Tail-r

isk

pre

dic

tion

Sort

ing

port

folios

Per

iod

TR�

KJ

tT

R�

BT

Xt

KJ

tB

TX

ta

t

10.0

7(1

.83)

0.0

1(0

.20)

0.0

2(0

.46)

0.0

9(1

.90)

One

month

20.1

0(2

.79)

0.0

9(1

.75)

0.0

8(1

.72)

0.1

1(2

.50)

Aver

age

�1.4

4(�

3.3

0)

30.1

3(3

.82)

0.1

4(2

.29)

0.1

2(3

.19)

0.1

5(3

.05)

FF3

�1.1

3(�

2.9

9)

40.1

1(3

.02)

0.1

3(2

.01)

0.1

5(3

.30)

0.1

5(3

.29)

FF3þ

MO

M�

1.1

7(�

3.0

2)

50.0

9(2

.37)

0.1

1(1

.97)

0.1

2(2

.50)

0.1

1(2

.40)

FF3þ

MO

Mþ

LIQ

�1.2

2(�

2.6

5)

60.1

0(2

.31)

0.1

1(1

.72)

0.1

1(2

.22)

0.0

8(1

.84)

FF3þ

MO

Mþ

LIQþ

VO

L�

0.7

9(�

1.7

6)

70.0

9(2

.07)

0.1

1(1

.54)

0.1

2(2

.49)

0.0

9(1

.91)

One

yea

r

80.0

5(1

.20)

0.0

4(0

.54)

0.1

2(2

.52)

0.0

7(1

.37)

Aver

age

�12.0

5(�

3.3

6)

90.0

4(0

.75)

0.0

2(0

.20)

0.0

9(2

.13)

0.0

2(0

.46)

FF3

�4.2

8(�

1.9

0)

10

0.0

4(0

.76)

0.0

3(0

.33)

0.0

8(1

.78)

0.0

1(0

.15)

FF3þ

MO

M�

4.6

7(�

2.3

1)

11

0.0

5(0

.82)

0.0

4(0

.56)

0.0

8(1

.91)

0.0

1(0

.22)

FF3þ

MO

Mþ

LIQ

�5.7

2(�

2.6

2)

12

0.0

3(0

.54)

0.0

4(0

.54)

0.0

9(1

.90)

0.0

1(0

.25)

FF3þ

MO

Mþ

LIQþ

VO

L�

3.1

5(�

0.5

7)

Fin

anci

alse

ctor

tail

risk

10.1

1(2

.22)

0.1

3(1

.55)

0.0

0(�

0.0

6)

0.0

7(1

.01)

One

month

20.1

4(2

.52)

0.1

5(1

.53)

0.0

7(0

.99)

0.1

2(2

.01)

Aver

age

�1.9

0(�

3.9

0)

30.1

6(2

.76)

0.1

7(1

.76)

0.1

5(2

.27)

0.1

8(3

.31)

FF3

�1.3

4(�

3.2

9)

40.1

1(1

.84)

0.1

4(1

.32)

0.1

3(2

.19)

0.1

4(2

.46)

FF3þ

MO

M�

1.4

0(�

3.3

4)

50.1

3(2

.10)

0.2

0(1

.92)

0.0

9(1

.54)

0.0

7(0

.99)

FF3þ

MO

Mþ

LIQ

�1.5

2(�

3.1

4)

60.1

2(1

.90)

0.2

0(1

.75)

0.1

1(2

.05)

0.0

8(1

.29)

FF3þ

MO

Mþ

LIQþ

VO

L�

0.8

1(�

1.9

6)

70.1

2(1

.89)

0.2

2(1

.84)

0.1

2(2

.37)

0.0

8(1

.28)

One

yea

r

80.1

1(1

.60)

0.2

1(1

.64)

0.1

1(2

.31)

0.0

6(0

.95)

Aver

age

�18.1

8(�

3.9

1)

(conti

nued

)

Almeida et al. j Nonparametric Tail Risk and the Macroeconomy 371

Downloaded from https://academic.oup.com/jfec/article-abstract/15/3/333/3072611by Universite de Montreal useron 19 March 2018

Tab

le12.

Co

nti

nu

ed

10

indust

ryport

folios

Ret

urn

pre

dic

tion

Tail-r

isk

pre

dic

tion

Sort

ing

port

folios

Per

iod

TR�

KJ

tT

R�

BT

Xt

KJ

tB

TX

ta

t

90.0

9(1

.15)

0.2

0(1

.39)

0.1

1(2

.08)

0.0

5(0

.77)

FF3

�7.4

5(�

2.1

9)

10

0.0

9(1

.07)

0.1

9(1

.27)

0.1

0(1

.63)

0.0

3(0

.42)

FF3þ

MO

M�

9.3

9(�

3.0

2)

11

0.0

8(0

.94)

0.1

7(1

.12)

0.1

0(1

.55)

0.0

2(0

.31)

FF3þ

MO

Mþ

LIQ

�10.6

7(�

3.3

2)

12

0.0

7(0

.87)

0.1

7(1

.16)

0.0

9(1

.37)

0.0

0(0

.06)

FF3þ

MO

Mþ

LIQþ

VO

L9.8

7(0

.87)

Rea

lse

ctor

tail

risk

10.1

1(3

.39)

0.0

8(1

.30)

0.0

6(1

.35)

0.1

5(3

.60)

One

month

20.1

4(3

.94)

0.1

1(1

.60)

0.1

3(2

.90)

0.1

9(4

.85)

Aver

age

�1.5

7(�

3.7

3)

30.1

6(4

.17)

0.1

6(2

.29)

0.1

8(3

.91)

0.2

3(5

.24)

FF3

�1.1

6(�

3.2

7)

40.1

3(3

.61)

0.1

5(2

.27)

0.1

7(3

.41)

0.2

0(4

.44)

FF3þ

MO

M�

1.2

1(�

3.3

0)

50.1

3(3

.40)

0.1

7(3

.14)

0.1

3(2

.62)

0.1

4(3

.31)

FF3þ

MO

Mþ

LIQ

�1.3

1(�

2.9

9)

60.1

1(2

.87)

0.1

5(2

.53)

0.1

3(2

.41)

0.1

3(3

.09)

FF3þ

MO

Mþ

LIQþ

VO

L�

0.9

0(�

2.1

9)

70.1

1(2

.91)

0.1

5(2

.34)

0.1

2(2

.36)

0.1

2(2

.82)

One

yea

r

80.0

8(2

.21)

0.1

0(1

.45)

0.1

2(2

.45)

0.1

1(2

.52)

Aver

age

�14.1

5(�

3.5

4)

90.0

6(1

.47)

0.0

7(0

.98)

0.1

1(2

.19)

0.0

8(1

.94)

FF3

�4.1

6(�

1.7

2)

10

0.0

6(1

.42)

0.0

8(1

.10)

0.0

8(1

.86)

0.0

5(1

.28)

FF3þ

MO

M�

4.6

6(�

2.1

2)

11

0.0

4(1

.06)

0.0

6(0

.88)

0.0

8(1

.94)

0.0

5(1

.39)

FF3þ

MO

Mþ

LIQ

�5.6

1(�

2.5

1)

12

0.0

4(1

.04)

0.0

7(0

.91)

0.0

7(1

.65)

0.0

4(0

.87)

FF3þ

MO

Mþ

LIQþ

VO

L7.6

2(0

.80)

Note

s:A

ssort

edro

bust

nes

ste

sts

are

run

for

ata

il-r

isk

mea

sure

const

ruct

edusi

ng

indust

ryport

folios.

Colu

mns

2–5

featu

reco

effici

ents

and

t-st

ats

of

pre

dic

tive

regre

ssio

ns

for

mark

et

retu

rns

for

two

esti

mati

ng

sam

ple

s,th

efirs

tone

corr

espondin

gto

Kel

lyand

Jiang

(2015,

KJ)

and

the

seco

nd

one

toB

oller

slev

,T

odoro

v,

and

Xu

(2015,

BT

X).

InC

olu

mns

6–9

we

report

the

coef

fici

ents

of

pre

dic

tive

regre

ssio

ns

of

the

KJ

and

BT

Xm

easu

res

by

our

tail-r

isk

mea

sure

.R

eturn

sfo

rth

ehig

h-m

inus-

low

sort

edport

foli

os

are

report

edin

Colu

mns

10–

12.A

llt-

stati

stic

sare

com

pute

dusi

ng

New

ey–W

est

standard

erro

rs.

372 Journal of Financial Econometrics

Downloaded from https://academic.oup.com/jfec/article-abstract/15/3/333/3072611by Universite de Montreal useron 19 March 2018

Tab

le13.

Ro

bu

stn

ess

:V

aR

tail

risk

Ret

urn

pre

dic

tion

Tail-r

isk

pre

dic

tion

Sort

ing

port

folios

Per

iod

TR�

KJ

tT

R�

BT

Xt

KJ

tB

TX

ta

t

10.0

5(0

.76)

�0.0

1(�

0.1

2)

�0.0

1(�

0.3

2)

0.2

7(1

.36)

One

month

20.0

6(0

.80)

�0.0

2(�

0.1

7)

0.0

1(0

.26)

0.1

2(0

.69)

Aver

age

�1.1

7(�

2.5

8)

30.0

8(1

.08)

0.0

3(0

.24)

0.0

2(0

.76)

0.2

2(0

.91)

FF3

�0.8

6(�

1.9

6)

40.0

9(1

.26)

0.0

8(0

.69)

0.0

3(0

.99)

0.1

1(0

.75)

FF3þ

MO

M�

0.8

4(�

1.8

9)

50.0

9(1

.29)

0.1

1(1

.10)

0.0

4(0

.97)

0.1

5(0

.84)

FF3þ

MO

Mþ

LIQ

�0.8

4(�

1.6

5)

60.0

9(1

.18)

0.1

2(1

.23)

0.0

5(1

.30)

0.1

4(0

.77)

FF3þ

MO

Mþ

LIQþ

VO

L0.0

6(0

.13)

70.0

9(1

.10)

0.1

5(1

.19)

0.0

7(1

.51)

0.1

5(0

.75)

One

yea

r

80.0

7(0

.79)

0.1

5(1

.02)

0.0

8(1

.64)

0.1

7(0

.67)

Aver

age

�10.9

7(�

3.2

4)

90.0

7(0

.73)

0.1

3(0

.90)

0.0

9(1

.83)

0.1

3(0

.70)

FF3

�2.1

6(�

0.7

7)

10

0.0

7(0

.75)

0.1

5(0

.98)

0.1

0(1

.78)

0.1

7(0

.87)

FF3þ

MO

M�

2.1

0(�

0.9

5)

11

0.0

6(0

.62)

0.1

9(1

.08)

0.1

4(2

.45)

0.1

6(0

.85)

FF3þ

MO

Mþ

LIQ

�1.8

3(�

0.7

5)

12

0.0

5(0

.50)

0.1

8(1

.02)

0.1

6(2

.86)

0.1

7(1

.20)

FF3þ

MO

Mþ

LIQþ

VO

L18.0

5(1

.63)

Note

s:A

ssort

edro

bust

nes

ste

sts

are

run

for

ari

sk-n

eutr

al

VaR

mea

sure

.C

olu

mns

2–5

featu

reco

effici

ents

and

t-st

ats

of

pre

dic

tive

regre

ssio

ns

for

mark

etre

turn

sfo

rtw

oes

tim

ati

ng

sam

ple

s,th

efirs

tone

corr

espondin

gto

Kel

lyand

Jiang

(2015,K

J)and

the

seco

nd

one

toB

oller

slev

,T

odoro

v,

and

Xu

(2015,B

TX

).In

Colu

mns

6–9

we

report

the

coef

fici

ents

of

pre

-

dic

tive

regre

ssio

ns

of

the

KJ

and

BT

Xm

easu

res

by

our

tail-r

isk

mea

sure

.R

eturn

sfo

rth

ehig

h-m

inus-

low

sort

edport

folios

are

report

edin

Colu

mns

10–12.

All

t-st

ati

stic

sare

com

-

pute

dusi

ng

New

ey–W

est

standard

erro

rs.

Almeida et al. j Nonparametric Tail Risk and the Macroeconomy 373

Downloaded from https://academic.oup.com/jfec/article-abstract/15/3/333/3072611by Universite de Montreal useron 19 March 2018

We can conclude that the risk-neutralized excess ES is a very robust measure of tail risk

once we aggregate the individual stock returns into portfolios irrespective of the nature of

the portfolios. Risk premia are of the same magnitude and estimated roughly with the same

degree of precision. The financial sector tail-risk measure tends to accentuate the premia

but the pattern remains the same.

4.3. Measuring Tail Risk from Risk-Neutral VaR

We have argued that ES should be preferred to VaR on theoretical grounds, but some

researchers have produced some empirical results that support the opposite. Therefore, we

construct a tail-risk measure based on the risk-neutral VaR that we used in computing our

excess ES for the base case with the five principal components of the 25 size and book-to-

market portfolios. We report some selected representative results in Table 13. Let us start

with our benchmark risk premium measure for the high-minus-low portfolio in the last col-

umns of the table. We can see right away that the only significant figures are the averages

when we do not control for other factors. Although the signs are preserved compared with

our base-case results in Table 4 when we control for other factors, the magnitudes are dif-

ferent and the statistical significance is either marginal or disappears.

The other columns illustrate also the poor performance of the risk-neutral VaR tail risk.

Market returns are not predicted well, neither in a long sample nor in a short one. No coef-

ficient is nearly significant. The same is true for the prediction of the Kelly and Jiang (2014)

tail risk or the option-based Bollerslev, Todorov, and Xu (2015) measure. These series were

predicted very well by the Hellinger excess ES measure, especially the Kelly Jiang measure.

This lack of predictive power extends to all series, except the macroeconomic uncertainty

index and the NBER indicator of recessions. Finally, impulse response functions for

employment and industrial production (not reported for space considerations) are essen-

tially zero.

5 Conclusion

We propose a new measure of tail risk based on the risk-neutralized returns of a few princi-

pal components of the cross-section of stock returns. We have shown that the results are

robust whether we compute these principal components from a set of aggregate portfolios

(Fama–French size and book-to-market portfolios or industry portfolios) or from the whole

cross-section of CRSP returns. This measure has two main advantages: it accounts for

changes in economic conditions through the risk-neutralization nonparametric procedure

and it is based only on stock returns. Not using option prices is useful to extend the sample

considerably but opens also the possibility to apply the measure in markets where liquid

option markets do not exist. Moreover, a similar measure can be computed for many other

asset markets.

The series computed from 1926 to the present captures all main events that have

increased uncertainty in the economy, whether they find their source in the financial market

or in the political arena. It is therefore a tail-risk measure that goes beyond financial sys-

temic risk. We have investigated thoroughly its relation with other tail-risk measures. For

the option-based measures, our empirical comparison revealed that the implications of

both measures were very similar in terms of risk premia for short and long holding periods.

However, we have shown that the nature of our series is very different from the tail-risk

374 Journal of Financial Econometrics

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measure proposed by Kelly and Jiang (2014). The latter is very persistent and does not cap-

ture the big uncertainty events. In terms of forecasting power of market returns our series

has a good performance in the short-run while the Kelly and Jiang (2014) tail risk predicts

well 6–12 months ahead.

Our extensive investigation of economic predictability has confirmed the usefulness of

our measure for capturing future conditions in employment, output, or more comprehen-

sive measures of real activity. A detailed comparison with the Bloom (2009) indicator of

economic uncertainty based on volatility has shown that our tail-risk measure adds new

information and implies different impulse response functions for employment and indus-

trial production. Finally, our measure predicts remarkably well the NBER recessions up to

12 months ahead and the data-rich aggregate financial and macroeconomic indicators of

economic uncertainty produced recently by Jurado, Ludvigson, and Ng (2015). Our

limited-information measure appears to be a very useful indicator to test theories of busi-

ness cycles with economic shocks or to estimate or calibrate equilibrium asset pricing mod-

els with disasters, ambiguity aversion, or disappointment aversion, all implying an

important role for tail events.

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